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PHILOSOPHY 


MATHEMATICS. 


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THE 


PHILOSOPHY 


OF 


MATHEMATICS, 


TRANSLATED   FROM   THE 


BY 

W.    M.    GILLESPIE,  LL.  D. 

PROFESSOR   OF    OIVII,   JtNOINEERINO   It  ADJ.    PKOF.    0»   MATHEMATICS 
IS   UNION    OOLtBaK. 


NEW    YORK: 
HARPER    &    BROTHERS,    PUBLISHERS, 

FRANKLIN     S  Q  U  A  R  K. 

1858. 


8  4  9  Q  4 

\J    J.   4/   «J    J. 


Entered,  according  to  Act  of  Congress,  in  the  year  one  thousand 
eight  hundred  and  fifty-one,  by 

HARPER  &  BROTHERS, 

in  the  Clerk's  Office  of  the  District  Court  of  the  Southern  District 
of  New  York. 


Engineering  & 

Mathematical 
Sciences 
Library 


PREFACE, 


THE  pleasure  and  profit  which  the  translator 
has  received  from  the  great  work  here  presented, 
have  induced  him  to  lay  it  before  his  fellow-teach- 
ers and  students  of  Mathematics  in  a  more  access- 
ible form  than  that  in  which  it  has  hitherto  ap- 
peared. The  want  of  a  comprehensive  map  of  the 
wide  region  of  mathematical  science — a  bird's-eye 
view  of  its  leading  features,  and  of  the  true  bear- 
ings and  relations  of  all  its  parts — is  felt  by  every 
thoughtful  student.  He  is  like  the  visitor  to  a 
great  city,  who  gets  no  just  idea  of  its  extent  and 
situation  till  he  has  seen  it  from  some  command- 
ing eminence.  To  have  a  panoramic  view  of  the 
whole  district — presenting  at  one  glance  all  the 
parts  in  due  co-ordination,  and  the  darkest  nooks 
clearly  shown — is  invaluable  to  either  traveller  or 
student.  It  is  this  which  has  been  most  perfect- 
ly accomplished  for  mathematical  science  by  the 
author  whose  work  is  here  presented. 

Clearness  and  depth,  comprehensiveness  and 
precision,  have  never,  perhaps,  been  so  remarkably 
united  as  in  AUGUSTE  COMTE.  He  views  his  sub- 
ject from  an  elevation  which  gives  to  each  part  of 
the  complex  whole  its  true  position  and  value, 
while  his  telescopic  glance  loses  none  of  the  need- 
ful details,  and  not  only  itself  pierces  to  the  heart 


yj  PREFACE. 

of  the  matter,  but  converts  its  opaqueness  into 
such  transparent  crystal,  that  other  eyes  are  en- 
abled to  see  as  deeply  into  it  as  his  own. 

Any  mathematician  who  peruses  this  volume 
will  need  no  other  justification  of  the  high  opin- 
ion here  expressed ;  but  others  may  appreciate  the 
following  endorsements  of  well-known  authorities. 
Mill,  in  his  "  Logic,"  calls  the  work  of  M.  Comte 
"  by  far  the  greatest  yet  produced  on  the  Philoso- 
phy of  the  sciences ;"  and  adds,  "  of  this  admira- 
ble work,  one  of  the  most  admirable  portions  is  that 
in  which  he  may  truly  be  said  to  have  created  the 
Philosophy  of  the  higher  Mathematics :"  Morell, 
in  his  "  Speculative  Philosophy  of  Europe,"  says, 
"  The  classification  given  of  the  sciences  at  large, 
and  their  regular  order  of  development,  is  unques- 
tionably a  master-piece  of  scientific  thinking,  as 
simple  as  it  is  comprehensive ;"  and  Lewes,  in 
his  "  Biographical  History  of  Philosophy,"  names 
Comte  "  the  Bacon  of  the  nineteenth  century," 
and  says,  "I  unhesitatingly  record  my  conviction 
that  this  is  the  greatest  work  of  our  age." 

The  complete  work  of  M.  Comte — his  "  Cours 
de  Philosophic  Positive" — fills  six  large  octavo  vol- 
umes, of  six  or  seven  hundred  pages  each,  two 
thirds  of  the  first  volume  comprising  the  purely 
mathematical  portion.  The  great  bulk  of  the 
"  Course"  is  the  probable  cause  of  the  fewness  of 
those  to  whom  even  this  section  of  it  is  known. 
Its  presentation  in  its  present  form  is  therefore  felt 
by  the  translator  to  be  a  most  useful  contribution 
to  mathematical  progress  in  this  country. 


PREFACE. 


Vll 


The  comprehensiveness  of  the  style  of  the  au- 
thor— grasping  all  possible  forms  of  an  idea  in  one 
Briarean  sentence,  armed  at  all  points  against 
leaving  any  opening  for  mistake  or  forgetfulness 
— occasionally  verges  upon  cumbersomeness  and 
formality.  The  translator  has,  therefore,  some- 
times taken  the  liberty  of  breaking  up  or  condens- 
ing a  long  sentence,  and  omitting  a  few  passages 
not  absolutely  necessary,  or  referring  to  the  pecu- 
liar "  Positive  philosophy"  of  the  author ;  but  he 
has  generally  aimed  at  a  conscientious  fidelity  to 
the  original.  It  has  often  been  difficult  to  retain 
its  fine  shades  and  subtile  distinctions  of  mean- 
ing, and,  at  the  same  time,  replace  the  peculiarly 
appropriate  French  idioms  by  corresponding  En- 
glish ones.  The  attempt,  however,  has  always 
been  made,  though,  when  the  best  course  has  been 
at  all  doubtful,  the  language  of  the  original  has 
been  followed  as  closely  as  possible,  and,  when 
necessary,  smoothness  and  grace  have  been  un- 
hesitatingly sacrificed  to  the  higher  attributes  of 
clearness  and  precision. 

Some  forms  of  expression  may  strike  the  reader 
as  unusual,  but  they  have  been  retained  because 
they  were  characteristic,  not  of  the  mere  language 
of  the  original,  but  of  its  spirit.  When  a  great 
thinker  has  clothed  his  conceptions  in  phrases 
which  are  singular  even  in  his  own  tongue,  he  who 
professes  to  translate  him  is  bound  faithfully  to 
preserve  such  forms  of  speech,  as  far  as  is  practi- 
cable ;  and  this  has  been  here  done  with  respect 
to  such  peculiarities  of  expression  as  belong  to  the 


PREFACE. 

author,  not  as  a  foreigner,  but  as  an  individual — 
not  "because  he  writes  in  French,  but  because  he 
is  Auguste  Comte. 

The  young  student  of  Mathematics  should  not 
attempt  to  read  the  whole  of  this  volume  at  once, 
but  should  peruse  each  portion  of  it  in  connexion 
with  the  temporary  subject  of  his  special  study : 
the  first  chapter  of  the  first  book,  for  example, 
while  he  is  studying  Algebra ;  the  first,  chapter  of 
the  second  book,  when  he  has  made  some  progress 
in  Geometry;  and  so  with  the  rest.  Passages 
which  are  obscure  at  the  first  reading  will  bright- 
en up  at  the  second ;  and  as  his  own  studies  cover 
a  larger  portion  of  the  field  of  Mathematics,  he 
will  see  more  and  more  clearly  their  relations  to 
one  another,  and  to  those  which  he  is  next  to  take 
up.  For  this  end  he  is  urgently  recommended  to 
obtain  a  perfect  familiarity  with  the  "  Analytical 
Table  of  Contents,"  which  maps  out  the  whole 
subject,  the  grand  divisions  of  which  are  also  in- 
dicated in  the  Tabular  View  facing  the  title-page. 
Corresponding  heads  will  be  found  in  the  body  of 
the  work,  the  principal  divisions  being  in  SMALL 
CAPITALS,  and  the  subdivisions  in  Italics.  For 
these  details  the  translator  alone  is  responsible. 


ANALYTICAL  TABLE  OF  CONTENTS. 


INTPuODUCTION. 

GENERAL  CONSIDERATIONS   ON   MATHEMATICAL  SCI- 
ENCE     17 

THE  OBJECT  OF  MATHEMATICS 18 

Measuring  Magnitudes 18 

Difficulties 19 

General  Method . 20 

Illustrations 21 

1 .  Falling  Bodies 21 

2.  Inaccessible  Distances 23 

3.  Astronomical  Facts 24 

TRUE  DEFINITION  OF  MATHEMATICS 25 

A  Science,  not  an  Art 25 

ITS  TWO  FUNDAMENTAL  DIVISIONS 26 

Their  different  Objects 27 

Their  different  Natures 29 

Concrete  Mathematics 31 

Geometry  and  Mechanics 32 

Abstract  Matliematics 33 

The  Calculus,  or  Analysis 33 

EXTENT  OF  ITS  FIELD 35 

Its  Universality 36 

Its  Limitations 37 


x    ANALYTICAL  TABLE  OF  CONTENTS. 
BOOK  I. 

ANALYSIS. 

CHAPTER  I. 

Page 

GENERAL  VIEW  OF  MATHEMATICAL  ANALYSIS  .  45 

THE  TRUE  IDEA  OF  AN  EQUATION 46 

Division  of  Functions  into  Abstract  and  Concrete  ....  47 

Enumeration  of  Abstract  Functions 50 

DIVISIONS  OF  THE  CALCULUS 53 

The  Calculus  of  Values,  or  Arithmetic 57 

Its  Extent 57 

Its  true  Nature 59 

The  Calculus  of  Functions 61 

Two  Modes  of  obtaining  Equations 61 

1.  By  the  Relations  between  the  given  Quantities  .  61 

2.  By  the  Relations  between  auxiliary  Quantities  .  .  64 
Corresponding  Divisions  of  the  Calculus  of  Functions .  67 

CHAPTER  II. 

ORDINARY  ANALYSIS  ;   OR,  ALGEBRA.  69 

Its  Object 69 

Classification  of  Equations 70 

ALGEBRAIC  EQUATIONS 71 

Their  Classification 71 

ALGEBRAIC  RESOLUTION  OF  EQUATIONS .  72 

Its  Limits 72 

General  Solution 72 

What  we  know  in  Algebra '. 74 

NUMERICAL  RESOLUTION  OF  EQUATIONS 75 

Its  limited  Usefulness 76 

Different  Divisions  of  the  two  Systems 78 

THE  THEORY  OF  EQUATIONS 79 

THE  METHOD  OF  INDETERMINATE  COEFFICIENTS 80 

IMAGINARY  QUANTITIES 81 

NEGATIVE  QUANTITIES 81 

THE  PRINCIPLE  OF  HOMOGENEITY  .  .  84 


ANALYTICAL  TABLE  OF  CONTENTS. 


CHAPTER  III. 

TRANSCENDENTAL  ANALYSIS :  Pago 

ITS  DIFFERENT  CONCEPTIONS 88 

Preliminary  Remarks 88 

Its  early  History 89 

METHOD  OF  LEIBNITZ 91 

Infinitely  small  Elements 91 

Examples  : 

•    1.  Tangents 93 

2.  Rectification  of  an  Arc 94 

3.  Quadrature  of  a  Curve 95 

4.  Velocity  in  variable  Motion , 95 

5.  Distribution  of  Heat 96 

Generality  of  the  Formulas 97 

Demonstration  of  the  Method 98 

Illustration  by  Tangents 102 

METHOD  OF  NEWTON 103 

Method  of  Limits   103 

Examples : 

1,  Tangents 104 

2.  Rectifications 105 

Fluxions  and  Fluents 106 

METHOD  OF  LAGE.ANGE 108 

Derived  Functions 108 

An  extension  of  ordinary  Analysis v 108 

Example :  Tangents 109 

Fundamental  Identity  of  the  three  Methods 110 

Their  comparative  Value 113 

That  of  Leibnitz 113 

That. of  Newton '. 115 

That. of  Lagrange 117 


ANALYTICAL  TABLE  OF  CONTENTS. 


CHAPTER   IV. 

Page 

THE  DIFFERENTIAL  AND  INTEGRAL  CALCULUS  .  120 

ITS  TWO  FUNDAMENTAL  DIVISIONS 120 

THEIR.  RELATIONS  TO  EACH  OTHER 121 

1.  Use  of  the  Differential  Calculus  as  preparatory  to 

that  of  the  Integral 123 

2.  Employment  of  the  Differential  Calculus  alone.  .  125 

3.  Employment  of  the  Integral  Calculus  alone  ....  125 

Three  Classes  of  Questions  hence  resulting  ...  126 

THE  DIFFERENTIAL  CALCULUS 127 

Two  Cases  :  Explicit  and  Implicit  Functions 127 

Two  sub-Cases  :  a  single  Variable  or  several ....  1 29 
Two  other  Cases  :  Functions  separate  or  combined  130 
Reduction  of  all  to  the  Differentiation  of  the  ten  ele- 
mentary Functions 131 

Transformation  of  derived  Functions  for  new  Variables  132 

Different  Orders  of  Differentiation 133 

Analytical  Applications 133 

THE  INTEGRAL  CALCULUS 135 

Its  fundamental  Division  :  Explicit  and  Implicit  Func- 
tions    135 

Subdivisions  :  a  single  Variable  or  several 136 

Calculus  of  partial  Differences 137 

Another  Subdivision :  different  Orders  of  Differentiation  138 

Another  equivalent  Distinction 140 

Quadratures 142 

Integration  of  Transcendental  Functions 143 

Integration  by  Parts 143 

Integration  of  Algebraic  Functions 143 

Singular  Solutions 144 

Definite  Integrals 146 

Prospects  of  the  Integral  Calculus 1 48 


ANALYTICAL   TABLE   OF   CONTENTS. 


CHAPTER  V.  pagc 

THE  CALCULUS  OF  VARIATIONS     151 

PROBLEMS  GIVING  RISE  TO  IT 151 

Ordinary  Questions  of  Maxima  and  Minima 151 

A  new  Class  of  Questions 152 

Solid  of  least  Resistance ;  Brachystochrone  ;  Isope- 

rimeters 153 

ANALYTICAL  NATURE  op  THESE  QUESTIONS 154 

METHODS  OF  THE  OLDER  GEOMETERS 155 

METHOD  OF  LAGRANGE 156 

Two  Classes  of  Questions 157 

1.  Absolute  Maxima  and  Minima 157 

Equations  of  Limits 159 

A  more  general  Consideration 159 

2.  Relative  Maxima  and  Minima 160 

Other  Applications  of  the  Method  of  Variations 162* 

ITS  RELATIONS  TO  THE  ORDINARY  CALCULUS  .  .163 


CHAPTER  VI. 

THE  CALCULUS  OF  FINITE  DIFFERENCES 167 

Its  general  Character 167 

Its  true  Nature 168 

GENERAL  THEORY  OF  SERIES 170 

Its  Identity  with  this  Calculus 172 

PERIODIC  OR  DISCONTINUOUS  FUNCTIONS 173 

APPLICATIONS  OF  THIS  CALCULUS 173 

Series 173 

Interpolation 173 

Approximate  Rectification,  &c 174 


Xjv        ANALYTICAL  TABLE  OF  CONTENTS. 


BOOK    II. 

GEOMETRY. 

CHAPTER  I. 

Page 

A  GENERAL  VIEW  OF  GEOMETRY 179 

The  true  Nature  of  Geometry 179 

Two  fundamental  Ideas 181 

1.  The  Idea  of  Space 181 

2.  Different  kinds  of  Extension 182 

THE  FINAL  OBJECT  OF  GEOMETRY 184 

Nature  of  Geometrical  Measurement 185 

Of  Surfaces  and  Volumes 185 

Of  curve  Lines 187 

Of  right  Lines 189 

THE  INFINITE  EXTENT  OF  ITS  FlELD 190 

Infinity  of  Lines .  190 

Infinity  of  Surfaces 191 

Infinity  of  Volumes 192 

Analytical  Invention  of  Curves,  &c 193 

EXPANSION  OF  ORIGINAL  DEFINITION 193 

Properties  of  Lines  and  Surfaces 1 95 

Necessity  of  their  Study 195 

1.  To  find  the  most  suitable  Property 195 

2.  To  pass  from  the  Concrete  to  the  Abstract 1 97 

Illustrations  : 

Orbits  of  the  Planets 198 

Figure  of  the  Earth 1 99 

THE  TWO  GENERAL  METHODS  OF  GEOMETRY .  202 

Their  fundamental  Difference 203 

1°.  Different  Questions  with  respect  to  the   same 

Figure 204 

2°.  Similar  Questions  with  respect  to  different  Figures  204 

Geometry  of  the  Ancients ' 204 

Geometry  of  the  Moderns 205 

Superiority  of  the  Modern 207 

The  Ancient  the  base  of  the  Modern  ..,                       .  209 


ANALYTICAL   TABLE   OF   CONTENTS.         xv 


CHAPTER  II. 

ANCIENT  OR  SYNTHETIC  GEOMETRY 

Page 

ITS  PROPER  EXTENT 212 

Lines  ;  Polygons  ;  Polyhedrons 212 

Not  to  be  farther  restricted 213 

Improper  Application  of  Analysis 214 

Attempted  Demonstrations  of  Axioms 216 

GEOMETRY  OF  THE  RIGHT  LINE 217 

GRAPHICAL  SOLUTIONS 218 

Descriptive  Geometry  ....'. 220 

ALGEBRAICAL  SOLUTIONS 224 

Trigonometry 225 

Two  Methods  of  introducing  Angles  226 

1.  By  Arcs 226 

2.  By  trigonometrical  Lines 226 

Advantages  of  the  latter 226 

Its  Division  of  trigonometrical  Questions 227 

1.  Relations  between  Angles  and  trigonometrical 
Lines 228 

2.  Relations  between  trigonometrical  Lines  and 
Sides 228 

Increase  of  trigonometrical  Lines 228 

Study  of  the  Relations  between  them 230 


XV1        ANALYTICAL   TABLE   OF   CONTENTS. 


CHAPTER  III. 

MODERN  OR  ANALYTICAL  GEOMETRY 

Page 

THE  ANALYTICAL  REPRESENTATION  OF  FIGURES 232 

Reduction  of  Figure  to  Position 233 

Determination  of  the  position  of  a  Point 234 

PLANE  CURVES 237 

Expression  of  Lines  by  Equations 237 

Expression  of  Equations  by  Lines 238 

Any  change  in  the  Line  changes  the  Equation 240 

Every  "  Definition"  of  a  Line  is  an  Equation 241 

Choice  of  Co-ordinates t, 245 

Two  different  points  of  View 245 

1.  Representation  of  Lines  by  Equations 246 

2.  Representation  of  Equations  by  Lines 246 

Superiority  of  the  rectilinear  System 248 

Advantages  of  perpendicular  Axes 249 

SURFACES 251 

Determination  of  a  Point  in  Space 251 

Expression  of  Surfaces  by  Equations 253 

Expression  of  Equations  by  Surfaces 253 

CURVES  IN  SPACE  255 

Imperfections  of  Analytical  Geometry 258 

Relatively  to  Geometry 258 

Relatively  to  Analysis 258 


THE 


PHILOSOPHY   OF  MATHEMATICS, 


INTRODUCTION. 

GENERAL    CONSIDERATIONS. 

ALTHOUGH  Mathematical  Science  is  the  most  ancient 
and  the  most  perfect  of  all,  yet  the  general  idea  which 
we  ought  to  form  of  it  has  not  yet  been  clearly  deter- 
mined. Its  definition  and  its  principal  divisions  have 
remained  till  now  vague  and  uncertain.  Indeed  the 
plural  name — "  The  Mathematics" — by  which  we  com- 
monly designate  it,  would  alone  suffice  to  indicate  the 
want  of  unity  in  the  common  conception  of  it. 

In  truth,  it  was  not  till  the  commencement  of  the  last 
century  that  the  different  fundamental  conceptions  which 
constitute  this  great  science  were  each  of  them  suffi- 
ciently developed  to  permit  the  true  spirit  of  the  whole 
to  manifest  itself  with  clearness.  Since  that  epoch  the 
attention  of  geometers  has  been  too  exclusively  absorbed 
by  the  special  perfecting  of  the  different  branches,  and 
by  the  application  which  they  have  made  of  them  to  the 
most  important  laws  of  the  universe,  to  allow  them  to 
give  due  attention  to  the  general  system  of  the  science. 

But  at  the  present  time  the  progress  of  the  special 
departments  is  no  longer  so  rapid  as  to  forbid  the  con- 
templation of  the  whole.  The  science  of  mathematics 

B 


18  MATHEMATICAL   SCIENCE. 

is  now  sufficiently  developed,  both  in  itself  and  as  to  its 
most  essential  application,  to  have  arrived  at  that  state 
of  consistency  in  which  we  ought  to  strive  to  arrange  its 
different  parts  in  a  single  system,  in  order  to  prepare  for 
new  advances.  We  may  even  observe  that  the  last  im- 
portant improvements  of  the  science  have  directly  paved 
the  way  for  this  important  philosophical  operation,  by  im- 
pressing on  its  principal  parts  a  character  of  unity  which 
did  not  previously  exist. 

To  form  a  just  idea  of  the  object  of  mathematical  sci- 
ence, we  may  start  from  the  indefinite  and  meaningless 
definition  of  it  usually  given,  in  calling  it  "The  science 
of  magnitudes"  or,  which  is  more  definite,  "The  sci- 
ence which  has  for  its  object  the  measurement  of  mag- 
nitudes.'''' Let  us  see  how  we  can  rise  from  this  rough 
sketch  (which  is  singularly  deficient  in  precision  and 
depth,  though,  at  bottom,  just)  to  a  veritable  definition, 
worthy  of  the  importance,  the  extent,  and  the  difficulty 
of  the  science. 

THE    OBJECT   OP    MATHEMATICS. 

Measuring  Magnitudes.  The  question  of  measur- 
ing a  magnitude  in  itself  presents  to  the  mind  no  othei 
idea  than  that  of  the  simple  direct  comparison  of  this 
magnitude  with  another  similar  magnitude,  supposed  to 
be  known,  which  it  takes  for  the  unit  of  comparison 
among  all  others  of  the  same  kind.  According  to  this 
definition,  then,  the  science  of  mathematics — vast  and 
profound  as  it  is  with  reason  reputed  to  be — instead  of 
being  an  immense  concatenation  of  prolonged  mental  la- 
bours, which  offer  inexhaustible  occupation  to  our  in- 
tellectual activity,  Would  seem  to  consist  of  a  simple 


ITS   OBJECT.  19 

series  of  mechanical  processes  for  obtaining  directly  the 
ratios  of  the  quantities  to  be  measured  to  those  by  which 
we  wish  to  measure  them,  by  the  aid  of  operations  of 
similar  character  to  the  superposition  of  lines,  as  prac- 
ticed by  the  carpenter  with  his  rule. 

The  error  of  this  definition  consists  in  presenting  as 
direct  an  object  which  is  almost  always,  on  the  contrary, 
very  indirect.  The  direct  measurement  of  a  magnitude, 
by  superposition  or  any  similar  process,  is  most  frequent- 
ly an  operation  quite  impossible  for  us  to  perform ;  so 
that  if  we  had  no  other  means  for  determining  magni- 
tudes than  direct  comparisons,  we  should  be  obliged  to  re- 
nounce the  knowledge  of  most  of  those  which  interest  us. 

Difficulties.  The  force  of  this  general  observation 
will  be  understood  if  we  limit  ourselves  to  consider  spe- 
cially the  particular  case  which  evidently  offers  the  most 
facility — that  of  the  measurement  of  one  straight  line 
by  another.  This  comparison,  which  is  certainly  the 
most  simple  which  we  can  conceive,  can  nevertheless 
scarcely  ever  be  effected  directly.  In  reflecting  on  the 
whole  of  the  conditions  necessary  to  render  a  line  sus- 
ceptible of  a  direct  measurement,  we  see  that  most  fre- 
quently they  cannot  be  all  fulfilled  at  the  same  time. 
The  first  and  the  most  palpable  of  these  conditions — 
that  of  being  able  to  pass  over  the  line  from  one  end  of 
it  to  the  other,  in  order  to  apply  the  unit  of  measurement 
to  its  whole  length — evidently  excludes  at  once  by  far  the 
greater  part  of  the  distances  which  interest  us  the  most ; 
in  the  first  place,  all  the  distances  between  the  celestial 
bodies,  or  from  any  one  of  them  to  the  earth  ;  and  then, 
too,  even  the  greater  number  of  terrestrial  distances,  which 
are  so  frequently  inaccessible.  But  even  if  this  first  con 


20  MATHEMATICAL   SCIENCE. 

dition  be  found  to  be  fulfilled,  it  is  still  farther  necessary 
that  the  length  be  neither  too  great  nor  too  small,  which 
would  render  a  direct  measurement  equally  impossible. 
The  line  must  also  be  suitably  situated  ;  for  let  it  be  one 
which  we  could  measure  with  the  greatest  facility,  if  it 
were  horizontal,  but  conceive  it  to  be  turned,  up  vertical- 
ly, and  it  becomes  impossible  to  measure  it. 

The  difficulties  which  we  have  indicated  in  reference 
to  measuring  lines,  exist  in  a  very  much  greater  degree 
in  the  measurement  of  surfaces,  volumes,  velocities,  times, 
forces,  &c.  It  is  this  fact  which  makes  necessary  the 
formation  of  mathematical  science,  as  we  are  going  to 
see ;  for  the  human  mind  has  been  compelled  to  re- 
nounce, in  almost  all  cases,  the  direct  measurement  of 
magnitudes,  and  to  seek  .to  determine  them  indirectly, 
and  it  is  thus  that  it  has  been  led  to  the  creation  of 
mathematics. 

General  Method.  The  general  method  which  is  con- 
stantly employed,  and  evidently  the  only  one  conceiva- 
ble, to  ascertain  magnitudes  which  do  not  admit  of  a  di- 
rect measurement,  consists  in  connecting  them  with  oth- 
ers which  are  susceptible  of  being  determined  immediate- 
ly, and  by  means  of  which  we  succeed  in  discovering 
the  first  through  the  relations  which  subsist  between 
the  two.  Such  is  the  precise  object  of  mathematical 
science  viewed  as  a  whole.  In  order  to  form  a  suffi- 
ciently extended  idea  of  it,  we  must  consider  that  this 
indirect  determination  of  magnitudes  may  be  indirect  in 
very  different  degrees.  In  a  great  number  of  cases, 
which  are  often  the  most  important,  the  magnitudes,  by 
means  of  which  the  principal  magnitudes  sought  are  to 
be  determined,  cannot  themselves  be  measured  directly, 


ITS    OBJECT.  21 

and  must  therefore,  in  their  turn,  become  the  subject  of  a 
similar  question,  and  so  on  ;  so  that  on  many  occasions 
the  human  mind  is  obliged  to  establish  a  long  series  of 
intermediates  between  the  system  of  unknown  magni- 
tudes which  are  the  final  objects  of  its  researches,  and 
the  system  of  magnitudes  susceptible  of  direct  measure- 
ment, by  whose  means  we  finally  determine  the  first, 
with  which  at  first  they  appear  to  have  no  connexion. 

Illustrations.  Some  examples  will  make  clear  any 
thing  which  may  seem  too  abstract  in  the  preceding 
generalities. 

1.  Falling  Bodies.  Let  us  consider,  in  the  first  place, 
a  natural  phenomenon,  very  simple,  indeed,  but  which 
may  nevertheless  give  rise  to  a  mathematical  question, 
really  existing,  and  susceptible  of  actual  applications — 
the  phenomenon  of  the  vertical  fall  of  heavy  bodies. 

The  mind  the  most  unused  to  mathematical  concep- 
tions, in  observing  this  phenomenon,  perceives  at  once 
that  the  two  quantities  which  it  presents — namely,  the 
height  from  which  a  body  has  fallen,  and  the  time  of  its 
fall — are  necessarily  connected  with  each  other,  since  they 
vary  together,  and  simultaneously  remain  fixed ;  or,  in 
the  language  of  geometers,  that  they  are  "functions"  of 
each  other.  The  phenomenon,  considered  under  this 
point  of  view,  gives  rise  then  to  a  mathematical  ques- 
tion, which  consists  in  substituting  for  the  direct  meas- 
urement of  one  of  these  two  magnitudes,  when  it  is  im- 
possible, the  measurement  of  the  other.  It  is  thus,  for 
example,  that  we  may  determine  indirectly  the  depth  of 
a  precipice,  by  merely  measuring  the  time  that  a  heavy 
body  would  occupy  in  falling  to  its  bottom,  and  by  suit- 
able procedures  this  inaccessible  depth  will  be  known 


22  MATHEMATICAL   SCIENCE. 

with  as  much  precision  as  if  it  was  a  horizontal  line 
placed  in  the  most  favourable  circumstances  for  easy  and 
exact  measurement.  On  other  occasions  it  is  the  height 
from  which  a  body  has  fallen  which  it  will  be  easy  to  as- 
certain, while  the  time  of  the  fall  could  not  be  observed 
directly  ;  then  the  same  phenomenon  would  give  rise  to 
the  inverse  question,  namely,  to  determine  the  time  from 
the  height ;  as,  for  example,  if  we  wished  to  ascertain 
what  would  be  the  duration  of  the  vertical  fall  of  a  body 
falling  from  the  moon  to  the  earth. 

In  this  example  the  mathematical  question  is  very  sim 
pie,  at  least  when  we  do  not  pay  attention  to  the  variation 
in  the  intensity  of  gravity,  or  the  resistance  of  the  fluid 
which  the  body  passes  through  in  its  fall.  But,  to  ex- 
tend the  question,  we  have  only  to  consider  the  same 
phenomenon  in  its  greatest  generality,  in  supposing  the 
fall  oblique,  and  in  taking  into  the  account  all  the  prin- 
cipal circumstances.  Then,  instead  of  offering  simply 
two  variable  quantities  connected  with  each  other  by  a 
relation  easy  to  follow,  the  phenomenon  will  present  a 
much  greater  number ;  namely,  the  space  traversed, 
whether  in  a  vertical  or  horizontal  direction  ;  the  time 
employed  in  traversing  it ;  the  velocity  of  the  body  at 
each  point  of  its  course ;  even  the  intensity  and  the 
direction  of  its  primitive  impulse,  which  may  also  be 
viewed  as  variables ;  and  finally,  in  certain  cases  (to 
take  every  thing  into  the  account),  the  resistance  of  the 
medium  and  the  intensity  of  gravity.  All  these  different 
quantities  will  be  connected  with  one  another,  in  such  a 
way  that  each  in  its  turn  may  be  indirectly  determined 
by  means  of  the  others  ;  and  this  will  present  as  many 
distinct  mathematical  questions  as  there  may  be  co-exist- 


ITS   OBJECT.  23 

ing  magnitudes  in  the  phenomenon  under  consideration. 
Such  a  very  slight  change  in  the  physical  conditions  of 
a  problem  may  cause  (as  in  the  above  example)  a  mathe- 
matical research,  at  first  very  elementary,  to  be  placed  at 
once  in  the  rank  of  the  most  difficult  questions,  whose 
complete  and  rigorous  solution  surpasses  as  yet  the  ut- 
most power  of  the  human  intellect. 

2.  Inaccessible  Distances.  Let  us  take  a  second  ex- 
ample from  geometrical  phenomena.  Let  it  be  proposed 
to  determine  a  distance  which  is  not  susceptible  of  direct 
measurement ;  it  will  be  generally  conceived  as  making 
part  of  a  figure^  or  certain  system  of  lines,  chosen  in 
such  a  way  that  all  its  other  parts  may  be  observed  di- 
rectly ;  thus,  in  the  case  which  is  most  simple,  and  to 
which  all  the  others  may  be  finally  reduced,  the  pro- 
posed distance  will  be  considered  as  belonging  to  a  trian- 
gle, in  which  we  can  determine  directly  either  another 
side  and  two  angles,  or  two  sides  and  one  angle.  Thence- 
forward, the  knowledge  of  the  desired  distance,  instead 
of  being  obtained  directly,  will  be  the  result  of  a  math- 
ematical calculation,  which  will  consist  in  deducing  it 
from  the  observed  elements  by  means  of  the  relation 
which  connects  it  with  them.  This  calculation  will  be- 
come successively  more  and  more  complicated,  if  the  parts 
which  we  have  supposed  to  be  known  cannot  themselves 
be  determined  (as  is  most  frequently  the  case)  except  in 
an  indirect  manner,  by  the  aid  of  new  auxiliary  systems, 
the  number  of  which,  in  great  operations  of  this  kind, 
finally  becomes  very  considerable.  The  distance  being 
once  determined,  the  knowledge  of  it  will  frequently  be 
sufficient  for  obtaining  new  quantities,  which  will  become 
the  subject  of  new  mathematical  questions.  Thus,  when 


24  MATHEMATICAL    SCIENCE. 

we  know  at  what  distance  any  object  is  situated,  the 
simple  observation  of  its  apparent  diameter  will  evident- 
ly permit  us  to  determine  indirectly  its  real  dimensions, 
however  inaccessible  it  may  be,  and,  by  a  series  of  an- 
alogous investigations,  its  surface,  its  volume,  even  its 
weight,  and  a  number  of  other  properties,  a  knowledge 
of  which  seemed  forbidden  to  us. 

3.  Astronomical  Facts.  It  is  by  such  calculations 
that  man  has  been  able  to  ascertain,  not  only  the  dis- 
tances from  the  planets  to  the  earth,  and,  consequently, 
from  each  other,  but  their  actual  magnitude,  their  true 
figure,  even  to  the  inequalities  of  their  surface ;  and,  what 
seemed  still  more  completely  hidden  from  us,  their  re- 
spective masses,  their  mean  densities,  the  principal  cir- 
cumstances of  the  fair  of  heavy  bodies  on  the  surface  of 
each  of  them,  &c. 

By  the  power  of  mathematical  theories,  all  these  dif- 
ferent results,  and  many  others  relative  to  the  different 
classes  of  mathematical  phenomena,  have  required  no 
other  direct  measurements  than  those  of  a  very  small 
number  of  straight  lines,  suitably  chosen,  and  of  a  great- 
er number  of  angles.  We  may  even  say,  with  perfect 
truth,  so  as  to  indicate  in  a  word  the  general  range  of 
the  science,  that  if  we  did  not  fear  to  multiply  calcula- 
tions unnecessarily,  and  if  we  had  not,  in  consequence, 
to  reserve  them  for  the  determination  of  the  quantities 
which  could  not  be  measured  directly,  the  determina- 
tion of  all  the  magnitudes  susceptible  of  precise  estima- 
tion, which  the  various  orders  of  phenomena  can  offer  us, 
could  be  finally  reduced  to  the  direct  measurement  of  a 
single  straight  line  and  of  a  suitable  number  of  angles. 


ITS    TRUE    DEFINITION.  25 

TRUE    DEFINITION    OP    MATHEMATICS. 

We  are  now  able  to  define  mathematical  science  with 
precision,  by  assigning  to  it  as  its  object  the  indirect 
measurement  of  magnitudes,  and  by  saying  it  constantly 
proposes  to  determine  certain  magnitudes  from  others 
by  means  of  the  precise  relations  existing  between  them. 

This  enunciation,  instead  of  giving  the  idea  of  only  an 
art,  as  do  all  the  ordinary  definitions,  characterizes  im- 
mediately a  true  science,  and  shows  it  at  once  to  be  com- 
posed of  an  immense  chain  of  intellectual  operations, 
which  may  evidently  become  very  complicated,  because 
of  the  series  of  intermediate  links  which  it  will  be  neces- 
sary to  establish  between  the  unknown  quantities  and 
those  which  admit  of  a  direct  measurement ;  of  the  num- 
ber of  variables  coexistent  in  the  proposed  question  ;  and 
of  the  nature  of  the  relations  between  all  these  different 
magnitudes  furnished  by  the  phenomena  under  consid- 
eration. According  to  such  a  definition,  the  spirit  of 
mathematics  consists  in  always  regarding  all  the  quan- 
tities which  any  phenomenon  can  present,  as  connected 
and  interwoven  with  one  another,  with  the  view  of  de- 
ducing them  from  one  another.  Now  there  is  evidently 
no  phenomenon  which  cannot  give  rise  to  considerations 
of  this  kind  ;  whence  results  the  naturally  indefinite  ex- 
tent and  even  the  rigorous  logical  universality  of  math- 
ematical science.  We  shall  seek  farther  on  to  circum- 
scribe as  exactly  as  possible  its  real  extension. 

The  preceding  explanations  establish  clearly  the  pro- 
priety of  the  name  employed  to  designate  the  science 
which  we  are  considering.  This  denomination,  which 
has  taken  to-day  so  definite  a  meaning  by  itself  signifies 


26  MATHEMATICAL    SCIENCE. 

simply  science  in  general.  Such  a  designation,  rigor- 
ously  exact  for  the  Greeks,  who  had  no  other  real  sci- 
ence, could  be  retained  by  the  moderns  only  to  indicate 
the  mathematics  as  the  science,  beyond  all  others — the 
science  of  sciences. 

Indeed,  every  true  science  has  for  its  object  the  de- 
termination of  certain  phenomena  by  means  of  others,  in 
accordance  with  the  relations  which  exist  between  them. 
Every  science  consists  in  the  co-ordination  of  facts ;  if 
the  different  observations  were  entirely  isolated,  there 
would  be  no  science.  We  may  even  say,  in  general  terms, 
that  science  is  essentially  destined  to  dispense,  so  far  as 
the  different  phenomena  permit  it,  with  all  direct  ob- 
servation, by  enabling  us  to  deduce  from  the  smallest 
possible  number  of  immediate  data  the  greatest  possible 
number  of  results.  Is  not  this  the  real  use,  whether  in 
speculation  or  in  action,  of  the  laws  which  we  succeed 
in  discovering  among  natural  phenomena  ?  Mathemat- 
ical science,  in  this  point  of  view,  merely  pushes  to  the 
highest  possible  degree  the  same  kind  of  researches  which 
are  pursued,  in  degrees  more  or  less  inferior,  by  every 
real  science  in  its  respective  sphere. 

ITS    TWO    FUNDAMENTAL    DIVISIONS. 

We  have  thus  far  viewed  mathematical  science  only 
as  a  whole,  without  paying  any  regard  to  its  divisions. 
We  must  now,  in  order  to  complete  this  general  view, 
and  to  form  a  just  idea  of  the  philosophical  character  of 
the  science,  consider  its  fundamental  division.  The  sec- 
ondary divisions  will  be  examined  in  the  following  chap- 
ters. 

This  principal  division,  which  we  are  about  to  investi- 


ITS    TWO  DIVISIONS.  27 

gate,  can  be  truly  rational,  and  derived  from  the  real  na- 
ture of  the  subject,  only  so  far  as  it  spontaneously  pre- 
sents itself  to  us,  in  making  the  exact  analysis  of  a  com- 
plete mathematical  question.  We  will,  therefore,  hav- 
ing determined  above  what  is  the  general  object  of  math- 
ematical labours,  now  characterize  with  precision  the 
principal  different  orders  of  inquiries,  of  which  they  are 
constantly  composed. 

Their  different  Objects.  The  complete  solution  of 
every  mathematical  question  divides  itself  necessarily 
into  two  parts,  of  natures  essentially  distinct,  and  with 
relations  invariably  determinate.  We  have  seen  that 
every  mathematical  inquiry  has  for  its  object  to  deter- 
mine unknown  magnitudes,  according  to  the  relations  be- 
tween them  and  known  magnitudes.  Now  for  this  ob- 
ject, it  is  evidently  necessary,  in  the  first  place,  to  as- 
certain with  precision  the  relations  which  exist  between 
the  quantities  which  we  are  considering.  This  first 
branch  of  inquiries  constitutes  that  which  I  call  the  con- 
crete part  of  the  solution.  When  it  is  finished,  the  ques- 
tion changes ;  it  is  now  reduced  to  a  pure  question  of 
numbers,  consisting  simply  -in  determining  unknown 
numbers,  when  we  know  what  precise  relations  connect 
them  with  known  numbers.  This  second  branch  of  in- 
quiries is  what  I  call  the  abstract  part  of  the  solution. 
Hence  follows  the  fundamental  division  of  general  math- 
ematical science  into  two  great  sciences — ABSTRACT  MATH- 
EMATICS, and  CONCRETE  MATHEMATICS. 

This  analysis  may  be  observed  in  every  complete 
mathematical  question,  however  simple  or  complicated 
it  may  be.  A  single  example  will  suffice  to  make  it 
intelligible. 


28  MATHEMATICAL    SCIENCE. 

Taking  up  again  the  phenomenon  of  the  vertical  fall 
of  a  heavy  body,  and  considering  the  simplest  case,  we 
see  that  in  order  to  succeed  in  determining,  by  means  of 
one  another,  the  height  whence  the  body  has  fallen,  and 
the  duration  of  its  fall,  we  must  commence  by  discovering 
the  exact  relation  of  these  two  quantities,  or,  to  use  the 
language  of  geometers,  the  equation  which  exists  be- 
tween them.  Before  this  first  research  is  completed, 
every  attempt  to  determine  numerically  the  value  of  one 
of  these  two  magnitudes  from  the  other  would  evidently 
be  premature,  for  it  would  have  no  basis.  It  is  not  enough 
to  know  vaguely  that  they  depend  on  one  another — which 
every  one  at  once  perceives — but  it  is  necessary  to  de- 
termine in  what  this  dependence  consists.  This  inquiry 
may  be  very  difficult,  and  in  fact,  in  the  present  case, 
constitutes  incomparably  the  greater  part  of  the  problem. 
The  true  scientific  spirit  is  so  modern,  that  no  one,  per- 
haps, before  Galileo,  had  ever  remarked  the  increase  of 
velocity  which  a  body  experiences  in  its  fall :  a  circum- 
stance which  excludes  the  hypothesis,  towards  which  our 
mind  (always  involuntarily  inclined  to  suppose  in  every 
phenomenon  the  most  simple  functions,  without  any  oth- 
er motive  than  its  greater  facility  in  conceiving  them) 
would  be  naturally  led,  that  the  height  was  proportion- 
al to  the  time.  In  a  word,  this  first  inquiry  terminated 
in  the  discovery  of  the  law  of  Galileo. 

When  this  concrete  part  is  completed,  the  inquiry  be- 
comes one  of  quite  another  nature.  Knowing  that  the 
spaces  passed  through  by  the  body  in  each  successive  sec- 
ond of  its  fall  increase  as  the  series  of  odd  numbers,  we 
have  then  a  problem  purely  numerical  and  abstract ;  to 
deduce  the  height  from  the  time,  or  the  time  from  the 


ITS    TWO   DIVISIONS.  £9 

height ;  and  this  consists  in  finding  that  the  first  of  these 
two  quantities,  according  to  the  law  which  has  been  es- 
tablished, is  a  known  multiple  of  the  second  power  of  the 
other ;  from  which,  finally,  we  have  to  calculate  the  value 
of  the  one  when  that  of  the  other  is  given. 

In  this  example  the  concrete  question  is  more  difficult 
than  the  abstract  one.  The  reverse  would  be  the  case 
if  we  considered  the  same  phenomenon  in  its  greatest 
generality,  as  I  have  done  above  for  another  object. 
According  to  the  circumstances,  sometimes  the  first, 
sometimes  the  second,  of  these  two  parts  will  constitute 
the  principal  difficulty  of  the  whole  question ;  for  the 
mathematical  law  of  the  phenomenon  may  be  very  sim- 
ple, but  very  difficult  to  obtain,  or  it  may  be  easy  to  dis- 
cover, but  very  complicated ;  so  that  the  two  great  sec- 
tions of  mathematical  science,  when  we  compare  them 
as  wholes,  must  be  regarded  as  exactly  equivalent  in  ex- 
tent and  in  difficulty,  as  well  as  in  importance,  as  we 
shall  show  farther  on,  in  considering  each  of  them  sep- 
arately. 

Their  different  Natures.  These  two  parts,  essentially 
distinct  in  their  object,  as  we  have  just  seen,  are  no  less 
so  with  regard  to  the  nature  of  the  inquiries  of  which 
they  are  composed. 

The  first  should  be  called  concrete,  since  it  evidently 
depends  on  the  character  of  the  phenomena  considered, 
and  must  necessarily  vary  when  we  examine  new  phe- 
nomena ;  while  the  second  is  completely  independent  of 
the  nature  of  the  objects  examined,  and  is  concerned  with 
only  the  numerical  relations  which  they  present,  for  which 
reason  it  should  be  called  abstract.  The  same  relations 
may  exist  in  a  great  number  of  different  phenomena, 


30  MATHEMATICAL   SCIENCE. 

which,  in  spito  of  their  extreme  diversity,  will  be  yiewed 
by  the  geometer  as  offering  an  analytical  question  sus- 
ceptible, when  studied  by  itself,  of  being  resolved  once 
for  all.  Thus,  for  instance,  the  same  law  which  exists 
between  the  space  and  the  time  of  the  vertical  fall  of  a 
body  in  a  vacuum,  is  found  again  in  many  other  phe- 
nomena which  offer  no  analogy  with  the  first  nor  with 
each  other  ;  for  it  expresses  the  relation  between  the  sur- 
face of  a  spherical  body  and  the  length  of  its  diameter  ; 
it  determines,  in  like  manner,  the  decrease  of  the  intensity 
of  light  or  of  heat  in  relation  to  the  distance  of  the  ob- 
jects lighted  or  heated,  &c.  The  abstract  part,  com- 
mon to  these  different  mathematical  questions,  having 
been  treated  in  reference  to  one  of  these,  will  thus  have 
been  treated  for  all ;  while  the  concrete  part  will  have 
necessarily  to  be  again  taken  up  for  each  question  sep- 
arately, without  the  solution  of  any  one  of  them  being 
able  to  give  any  direct  aid,  in  that  connexion,  for  the  so- 
lution of  the  rest. 

The  abstract  part  of  mathematics  is,  then,  general  in 
its  nature  ;  the  concrete  part,  special. 

To  present  this  comparison  under  a  new  point  of  view, 
we  may  say  concrete  mathematics  has  a  philosophical 
character,  which  is  essentially  experimental,  physical, 
phenomenal ;  while  that  of  abstract  mathematics  is  pure- 
ly logical,  rational.  The  concrete  part  of  every  mathe- 
matical question  is  necessarily  founded  on  the  considera- 
tion of  the  external  world,  and  could  never  be  resolved 
by  a  simple  series  of  intellectual  combinations.  The  ab- 
stract part,  on  the  contrary,  when  it  has  been  very  com- 
pletely separated,  can  consist  only  of  a  series  of  logical 
deductions,  more  or  less  prolonged  ;  for  if  we  have  once 


CONCRETE    MATHEMATICS  ^1 

found  the  equations  of  a  phenomenon,  the  determination 
of  the  quantities  therein  considered,  by  means  of  one  an- 
other, is  a  matter  for  reasoning  only,  whatever  the  diffi- 
culties may  be.  It  belongs  to  the  understanding  alone 
to  deduce  from  these  equations  results  which  are  evi- 
dently contained  in  them,  although  perhaps  in  a  very  in- 
volved manner,  without  there  being  occasion  to  consult 
anew  the  external  world ;  the  consideration  of  which, 
having  become  thenceforth  foreign  to  the  subject,  ought 
even  to  be  carefully  set  aside  in  order  to  reduce  the  la- 
bour to  its  true  peculiar  difficulty.  The  abstract  part 
of  mathematics  is  then  purely  instrumental,  and  is  only 
an  immense  and  admirable  extension  of  natural  logic  to  a 
certain  class  of  deductions.  On  the  other  hand,  geome- 
try and  mechanics,  which,  as  we  shall  see  presently,  con- 
stitute the  concrete  part,  must  be  viewed  as  real  natu- 
ral sciences,  founded  on  observation,  like  all  the  rest, 
although  the  extreme  simplicity  of  their  phenomena  per- 
mits an  infinitely  greater  degree  of  systematization, 
which  has  sometimes  caused  a  misconception  of  the  ex- 
perimental character  of  their  first  principles.  9 

We  see,  by  this  brief  general  comparison,  how  natural 
and  profound  is  our  fundamental  division  of  mathemati- 
cal science. 

We  have  now  to  circumscribe,  as  exactly  as  we  can 
in  this  first  sketch,  each  of  these  two  great  sections. 

CONCRETE   MATHEMATICS. 

Concrete  Mathematics  having  for  its  object  the  dis- 
covery of  the  equations  of  phenomena,  it  would  seem  at 
first  that  it  must  be  composed  of  as  many  distinct  sci- 
ences as  we  find  really  distinct  categories  among  natural 


32  MATHEMATICAL  SCIENCE 

phenomena.  But  we  are  yet  very  far  from  having  dis- 
covered mathematical  laws  in  all  kinds  of  phenomena ; 
we  shall  even  see,  presently,  that  the  greater  part  will 
very  probably  always  hide  themselves  from  our  investiga- 
tions. In  reality,  in  the  present  condition  of  the  human 
mind,  there  are  directly  but  two  great  general  classes  of 
phenomena,  whose  equations  we  constantly  know  ;  these 
are,  firstly,  geometrical,  and,  secondly,  mechanical  phe- 
nomena. Thus,  then,  the  concrete  part  of  mathematics 
is  composed  of  GEOMETRY  and  RATIONAL  MECHANICS. 

This  is  sufficient,  it  is  true,  to  give  to  it  a  complete 
character  of  logical  universality,  when  we  consider  all 
phenomena  from  the  most  elevated  point  of  view  of  nat- 
ural philosophy.  In  fact,  if  all  the  parts  of  the  universe 
were  conceived  as  immovable,  we  should  evidently  have 
only  geometrical  phenomena  to  observe,  since  all  would 
be  reduced  to  relations  of  form,  magnitude,  and  position; 
then,  having  regard  to  the  motions  which  take  place  in  it, 
we  would  have  also  to  consider  mechanical  phenomena. 
Hence  the  universe,  in  the  statical  point  of  view,  pre- 
sents onlyf geometrical  phenomena;  and,  considered  dy- 
namically, only  mechanical  phenomena.  Thus  geometry 
and  mechanics  constitute  the  two  fundamental  natural 
sciences,  in  this  sense,  that  all  natural  effects  may  be  con- 
ceived as  simple  necessary  results,  either  of  the  laws  of 
extension  or  of  the  laws  of  motion. 

But  although  this  conception  is  always  logically  pos- 
sible, the  difficulty  is  to  specialize  it  with  the  necessary 
precision,  and  to  follow  it  exactly  in  each  of  the  general 
cases  offered  to  us  by  the  study  of  nature ;  that  is,  to 
effectually  reduce  each  principal  question  of  natural  phi- 
losophy, for  a  certain  determinate  order  of  phenomena,  to 


ABSTRACT  MATHEMATICS.  33 

the  question  of  geometry  or  mechanics,  to  which  we  might 
rationally  suppose  it  should  be  brought.  This  transform- 
ation, which  requires  great  progress  to  have  been  previous- 
ly made  in  the  study  of  each  class  of  phenomena,  has  thus 
far  been  really  executed  only  for  those  of  astronomy,  and 
for  a  part  of  those  considered  by  terrestrial  physics,  prop- 
erly so  called.  It  is  thus  that  astronomy,  acoustics,  op- 
tics, &c.,  have  finally  become  applications  of  mathemat- 
ical science  to  certain  orders  of  observations.^  But  these 
applications  not  being  by  their  nature  rigorously  circum- 
scribed, to  confound  them  with  the  science  would  be  to 
assign  to  it  a  vague  and  indefinite  domain  ;  and  this  is 
done  in  the  usual  division,  so  faulty  in  so  many  other 
respects,  of  the  mathematics  into  "Pure"  and  "Ap- 
plied." 

ABSTRACT   MATHEMATICS. 

The  nature  of  abstract  mathematics  (the  general  divis- 
ion of  which  will  be  examined  in  the  following  chapter)  is 
clearly  and  exactly  determined.  It  is  composed  of  what  is 
called  the  Calculus,^  taking  this  word  in  its  greatest  ex- 
tent, which  reaches  from  the  most  simple  numerical  ope- 
rations to  the  most  sublime  combinations  of  transcendental 
analysis.  The  Calculus  has  the  solution  of  all  questions 


*  The  investigation  of  the  mathematical  phenomena  of  the  laws  of  heat 
by  Baron  Fourier  has  led  to  the  establishment,  in  an  entirely  direct  manner, 
of  Thermological  equations.  This  great  discovery  tends  to  elevate  our  phil- 
osophical hopes  as  to  the  future  extensions  of  the  legitimate  applications  of 
mathematical  analysis,  and  renders  it  proper,  in  the  opinion  of  the  author, 
to  regard  Thermology  as  a  third  principal  branch  of  concrete  mathematics. 

t  The  translator  has  felt  justified  in  employing  this  very  convenient  word 
(for  which  our  language  has  no  precise  equivalent)  as  an  English  one,  in  its 
most  extended  sense,  in  spite  of  its  being  often  popularly  confounded  with 
its  Differential  and  Integral  department. 

c 


34  MATHEMATICAL   SCIENCE. 

relating  to  numbers  for  its  peculiar  object.  Its  starting 
point  is,  constantly  and  necessarily,  the  knowledge  of  the 
precise  relations,  i.  e.,  of  the  equations,  between  the  dif- 
ferent magnitudes  which  are  simultaneously  considered  ; 
that  which  is,  on  the  contrary,  the  stopping-  point  of  con- 
crete mathematics.  However  complicated,  or  however  in- 
direct these  relations  may  be,  the  final  object  of  the  cal- 
culus always  is  to  obtain  from  them  the  values  of  the  un- 
known quantities  by  means  of  those  which  are  known. 
This  science,  although  nearer  perfection  than  any  other, 
is  really  little  advanced  as  yet,  so  that  this  object  is  rare- 
ly attained  in  a  manner  completely  satisfactory. 

Mathematical  analysis  is,  then,  the  true  rational  basis 
of  the  entire  system  of  our  actual  knowledge.  It  con- 
stitutes the  first  and  the  most  perfect  of  all  the  funda- 
mental sciences.  The  ideas  with  which  it  occupies  it- 
self are  the  most  universal,  the  most  abstract,  and  the 
most  simple  which  it  is  possible  for  us  to  conceive. 

This  peculiar  nature  of  mathematical  analysis  enables 
us  easily  to  explain  why,  when  it  is  properly  employed, 
it  is  such  a  powerful  instrument,  not  only  to  give  more 
precision  to  our  real  knowledge,  which  is  self-evident,  but 
especially  to  establish  an  infinitely  more  perfect  co-ordi- 
nation in  the  study  of  the  phenomena  which  admit  of 
that  application  ;  for,  our  conceptions  having  been  so 
generalized  and  simplified  that  a  single  analytical  ques- 
tion, abstractly  resolved,  contains  the  implicit  solution 
of  a  great  number  of  diverse  physical  questions,  the  hu- 
man mind  must  necessarily  acquire  by  these  means  a 
greater  facility  in  perceiving  relations  between  phenom- 
ena which  at  first  appeared  entirely  distinct  from  one 
another.  We  thus  naturally  see  arise,  through  the  me- 


ITS    EXTENT.  35 

dium  of  analysis,  the  most  frequent  and  the  most  unex- 
pected approximations  between  problems  which  at  first 
offered  no  apparent  connection,  and  which  we  often  end 
in  viewing  as  identical.  Could  we,  for  example,  with- 
out the  aid  of  analysis,  perceive  the  least  resemblance 
between  the  determination  of  the  direction  of  a  curve  at 
each  of  its  points  and  that  of  the  velocity  acquired  by  a 
body  at  every  instant  of  its  variable  motion  ?  and  yet 
these  questions,  however  different  they  may  be,  compose 
but  one  in  the  eyes  of  the  geometer. 

The  high  relative  perfection  of  mathematical  analysis 
is  as  easily  perceptible.  This  perfection  is  not  due,  as 
some  have  thought,  to  the  nature  of  the  signs  which  are 
employed  as  instruments  of  reasoning,  eminently  concise 
and  general  as  they  are.  In  reality,  all  great  analytical 
ideas  have  been  formed  without  the  algebraic  signs  hav- 
ing been  of  any  essential  aid,  except  for  working  them 
out  after  the  mind  had  conceived  them.  The  superior 
perfection  of  the  science  of  the  calculus  is  due  princi- 
pally to  the  extreme  simplicity  of  the  ideas  which  it  con- 
siders, by  whatever  signs  they  may  be  expressed  ;  so  that 
there  is  not  the  least  hope,  by  any  artifice  of  scientific 
language,  of  perfecting  to  the  same  degree  theories  which 
refer  to  more  complex  subjects,  and  which  are  necessarily 
condemned  by  their  nature  to  a  greater  or  less  logical  in- 
feriority. 

THE    EXTENT    OF    ITS    FIELD. 

Our  examination  of  the  philosophical  character  of  math- 
ematical science  would  remain  incomplete,  if,  after  hav- 
ing viewed  its  object  and  composition,  we  did  not  exam- 
ine the  real  extent  of  its  domain. 


36  MATHEMATICAL    SCIENCE. 

Its  Universality.  For  this  purpose  it  is  indispensa- 
ble to  perceive,  first  of  all,  that,  in  the  purely  logical 
point  of  view,  this  science  is  by  itself  necessarily  and 
rigorously  universal ;  for  there  is  no  question  whatever 
which  may  not  be  finally  conceived  as  consisting  in  de- 
termining certain  quantities  from  others  by  means  of  cer- 
tain relations,  and  consequently  as  admitting  of  reduc- 
tion, in  final  analysis,  to  a  simple  question  of  numbers. 
In  all  our  researches,  indeed,  on  whatever  subject,  our 
object  is  to  arrive  at  numbers,  at  quantities,  though  often 
in  a  very  imperfect  manner  and  by  very  uncertain  meth- 
ods. Thus,  taking  an  example  in  the  class  of  subjects 
the  least  accessible  to  mathematics,  the  phenomena  of 
living  bodies,  even  when  considered  (to  take  the  most 
complicated  case)  in  the  state  of  disease,  is  it  not  mani- 
fest that  all  the  questions  of  therapeutics  may  be  viewed 
as  consisting  in  determining  the  quantities  of  the  differ- 
ent agents  which  modify  the  organism,  and  which  must 
act  upon  it  to  bring  it  to  its  normal  state,  admitting,  for 
some  of  these  quantities  in  certain  cases,  values  which 
are  equal  to  zero,  or  negative,  or  even  contradictory  ? 

The  fundamental  idea  of  Descartes  on  the  relation  of 
the  concrete  to  the  abstract  in  mathematics,  has  proven, 
in  opposition  to  the  superficial  distinction  of  metaphys- 
ics, that  all  ideas  of  quality  may  be  reduced  to  those  of 
quantity.  This  conception,  established  at  first  by  its 
immortal  author  in  relation  to  geometrical  phenomena 
only,  has  since  been  effectually  extended  to  mechanical 
phenomena,  and  in  our  days  to  those  of  heat.  As  a  re- 
sult of  this  gradual  generalization,  there  are  now  no  ge- 
ometers who  do  not  consider  it,  in  a  purely  theoretical 
sense,  as  capable  of  being  applied  to  all  our  real  ideas  of 


ITS   EXTENT.  37 

every  sort,  so  that  every  phenomenon  is  logically  suscep- 
tible of  being  represented  by  an  equation ;  as  much  so, 
indeed,  as  is  a  curve  or  a  motion,  excepting  the  diffi- 
sulty  of  discovering  it,  and  then  of  resolving  it,  which 
may  be,  and  oftentimes  are,  superior  to  the  greatest  pow- 
ers of  the  human  mind. 

Its  Limitations.  Important  as  it  is  to  comprehend 
the  rigorous  universality,  in  a  logical  point  of  view,  of 
mathematical  science,  it  is  no  less  indispensable  to  con- 
sider now  the  great  real  limitations,  which,  through  the 
feebleness  of  our  intellect,  narrow  in  a  remarkable  de- 
gree its  actual  domain,  in  proportion  as  phenomena,  in 
becoming  special,  become  complicated. 

Every  question  may  be  conceived  as  capable  of  being 
reduced  to  a  pure  question  of  numbers ;  but  the  diffi- 
culty of  effecting  such  a  transformation  increases  so  much 
with  the  complication  of  the  phenomena  of  natural  phi- 
losophy, that  it  soon  becomes  insurmountable. 

This  will  be  easily  seen,  if  we  consider  that  to  bring 
a  question  within  the  field  of  mathematical  analysis,  we 
must  first  have  discovered  the  precise  relations  which  ex- 
ist between  the  quantities  which  are  found  in  the  phe- 
nomenon under  examination,  the  establishment  of  these 
equations  being  the  necessary  starting  point  of  all  ana- 
lytical labours.  This  must  evidently  be  so  much  the 
more  difficult  as  we  have  to  do  with  phenomena  which 
are  more  special,  and  therefore  more  complicated.  We 
shall  thus  find  that  it  is  only  in  inorganic  physics,  at 
the  most,  that  we  can  justly  hope  ever  to  obtain  that 
high  degree  of  scientific  perfection. 

The  first  condition  which  is  necessary  in  order  that 
phenomena  may  admit  of  mathematical  laws,  susceptible 

84994 


38  MATHEMATICAL  SCIENCE. 

of  being  discovered,  evidently  is,  that  their  different  quan- 
tities should  admit  of  being  expressed  by  fixed  numbers. 
We  soon  find  that  in  this  respect  the  whole  of  organic 
physics,  and  probably  also  the  most  complicated  parts  of 
inorganic  physics,  are  necessarily  inaccessible,  by  their 
nature,  to  our  mathematical  analysis,  by  reason  of  the 
extreme  numerical  variability  of  the  corresponding  phe- 
nomena. Every  precise  idea  of  fixed  numbers  is  truly 
out  of  place  in  the  phenomena  of  living  bodies,  when  we 
wish  to  employ  it  otherwise  than  as  a  means  of  relieving 
the  attention,  and  when  we  attach  any  importance  to  the 
exact  relations  of  the  values  assigned. 

We  ought  not,  however,  on  this  account,  to  cease  to 
conceive  all  phenomena  as  being  necessarily  subject  to 
mathematical  laws,  which  we  are  condemned  to  be  igno- 
rant of,  only  because  of  the  too  great  complication  of  the 
phenomena.  The  most  complex  phenomena  of  living 
bodies  are  doubtless  essentially  of  no  other  special  nature 
than  the  simplest  phenomena  of  unorganized  matter.  If 
it  were  possible  to  isolate  rigorously  each  of  the  simple 
causes  which  concur  in  producing  a  single  physiological 
phenomenon,  every  thing  leads  us  to  believe  that  it  would 
show  itself  endowed,  in  determinate  circumstances,  with 
a  kind  of  influence  and  with  a  quantity  of  action  as  ex- 
actly fixed  as  we  see  it  in  universal  gravitation,  a  veri- 
table type  of  the  fundamental  laws  of  nature. 

There  is  a  second  reason  why  we  cannot  bring  compli- 
cated phenomena  under  the  dominion  of  mathematical 
analysis.  Even  if  we  could  ascertain  the  mathematical 
law  which  governs  each  agent,  taken  by  itself,  the  com- 
bination of  so  great  a  number  of  conditions  would  render 
the  corresponding  mathematical  problem  so  far  above  our 


ITS  EXTENT.  39 

feeble  means,  that  the  question  would  remain  in  most 
cases  incapable  of  solution. 

To  appreciate  this  difficulty,  let  us  consider  how  com- 
plicated mathematical  questions  become,  even  those  relat- 
ing to  the  most  simple  phenomena  of  unorganized  bodies, 
when  we  desire  to  bring  sufficiently  near  together  the  ab- 
stract and  the  concrete  state,  having  regard  to  all  the 
principal  conditions  which  can  exercise  a  real  influence 
over  the  effect  produced.  We  know,  for  example,  that 
the  very  simple  phenomenon  of  the  flow  of  a  fluid  through 
a  given  orifice,  by  virtue  of  its  gravity  alone,  has  not  as 
yet  any  complete  mathematical  solution,  when  we  take 
into  the  account  all  the  essential  circumstances.  It  is 
the  same  even  with  the  still  more  simple  motion  of  a 
solid  projectile  in  a  resisting  medium. 

Why  has  mathematical  analysis  been  able  to  adapt  itself 
with  such  admirable  success  to  the  most  profound  study 
of  celestial  phenomena?  Because  they  are,  in  spite  of 
popular  appearances,  much  more  simple  than  any  others. 
The  most  complicated  problem  which  they  present,  that 
of  the  modification  produced  in  the  motions  of  two  bodies 
tending  towards  each  other  by  virtue  of  their  gravitation, 
by  the  influence  of  a  third  body  acting  on  both  of  them 
in  the  same  manner,  is  much  less  complex  than  the  most 
simple  terrestrial  problem.  And,  nevertheless,  even  it 
presents  difficulties  so  great  that  we  yet  possess  only 
approximate  solutions  of  it.  It  is  even  easy  to  see  that 
the  high  perfection  to  which  solar  astronomy  has  been 
able  to  elevate  itself  by  the  employment  of  mathematical 
science  is,  besides,  essentially  due  to  our  having  skilfully 
profited  by  all  the  particular,  and,  so  to  say,  accidental 
facilities  presented  by  the  peculiarly  favourable  consti- 


40  MATHEMATICAL  SCIENCE. 

'^V 

tution  of  our  planetary  system.  The  planets  which  com- 
pose it  are  quite  few  in  number,  and  their  masses  are  in 
general  very  unequal,  and  much  less  than  that  of  the 
sun ;  they  are,  besides,  very  distant  from  one  another ; 
they  have  forms  almost  spherical ;  their  orbits  are  nearly 
circular,  and  only  slightly  inclined  to  each  other,  and  so 
on.  It  results  from  all  these  circumstances  that  the  per- 
turbations are  generally  inconsiderable,  and  that  to  cal- 
culate them  it  is  usually  sufficient  to  take  into  the  ac- 
count, in  connexion  with  the  action  of  the  sun  on  each 
particular  planet,  the  influence  of  only  one  other  planet, 
capable,  by  its  size  and  its  proximity,  of  causing  percept- 
ible derangements. 

If,  however,  instead  of  such  a  state  of  things,  our  so- 
lar system  had  been  composed  of  a  greater  number  of 
planets  concentrated  into  a  less  space,  and  nearly  equal 
in  mass;  if  their  orbits  had  presented  very  different  in- 
clinations, and  considerable  eccentricities  ;  if  these  bodies 
had  been  of  a  more  complicated  form,  such  as  very  ec- 
centric ellipsoids,  it  is  certain  that,  supposing  the  same 
law  of  gravitation  to  exist,  we  should  not  yet  have  suc- 
ceeded in  subjecting  the  study  of  the  celestial  phenome- 
na to  our  mathematical  analysis,  and  probably  we  should 
not  even  have  been  able  to  disentangle  the  present  prin- 
cipal law. 

These  hypothetical  conditions  would  find  themselves 
exactly  realized  in  the  highest  degree  in  chemical  phe- 
nomena, if  we  attempted  to  calculate  them  by  the  theory 
of  general  gravitation. 

On  properly  weighing  the  preceding  considerations, 
the  reader  will  be  convinced,  I  think,  that  in  reducing 
the  future  extension  of  the  great  applications  of  mathe- 


ITS   EXTENT.  41 

matical  analysis,  which  are  really  possible,  to  the  field 
comprised  in  the  different  departments  of  inorganic  phys- 
ics, I  have  rather  exaggerated  than  contracted  the  ex- 
tent of  its  actual  domain.  Important  as  it  was  to  ren- 
der apparent  the  rigorous  logical  universality  of  mathe- 
matical science,  it  was  equally  so  to  indicate  the  condi- 
tions which  limit  for  us  its  real  extension,  so  as  not  to 
contribute  to  lead  the  human  mind  astray  from  the  true 
scientific  direction  in  the  study  of  the  most  complicated 
phenomena,  by  the  chimerical  search  after  an  impossible 
perfection. 

Having  thus  exhibited  the  essential  object  and  the 
principal  composition  of  mathematical  science,  as  well  as 
its  general  relations  with  the  whole  body  of  natural  phi- 
losophy, we  have  now  to  pass  to  the  special  examination 
of  the  great  sciences  of  which  it  is  composed. 


ANALYSIS  and  GEOMETRY  are  the  two  great  heads  under  which 
the  subject  is  about  to  be  examined.  To  these  M.  Comte  adds  Rational 
MECHANICS  ;  but  as  it  is  not  comprised  in  the  usual  idea  of  Mathematics, 
and  as  its  discussion  would  be  of  but  limited  utility  and  interest,  it  is  not 
included  in  the  present  translation. 


BOOK  I. 


ANALYSIS. 


BOOK   I. 
ANALYSIS. 


CHAPTER   I. 

GENERAL   VIEW   OP   MATHEMATICAL   ANALYSIS. 

IN  the  historical  development  of  mathematical  science 
since  the  time  of  Descartes,  the  advances  of  its  abstract 
portion  have  always  been  determined  by  those  of  its  con- 
crete portion ;  but  it  is  none  the  less  necessary,  in  or- 
der to  conceive  the  science  in  a  manner  truly  logical,  to 
consider  the  Calculus  in  all  its  principal  branches  before 
proceeding  to  the  philosophical  study  of  Geometry  and 
Mechanics.  Its  analytical  theories,  more  simple  and 
more  general  than  those  of  concrete  mathematics,  are  in 
themselves  essentially  independent  of  the  latter ;  while 
these,  on  the  contrary,  have,  by  their  nature,  a  continual 
need  of  the  former,  without  the  aid  of  which  they  could 
make  scarcely  any  progress.  Although  the  principal 
conceptions  of  analysis  retain  at  present  some  very  per- 
ceptible traces  of  their  geometrical  or  mechanical  origin, 
they  are  now,  however,  mainly  freed  from  that  primitive 
character,  which  no  longer  manifests  itself  except  in  some 
secondary  points ;  so  that  it  is  possible  (especially  since 
the  labours  of  Lagrange)  to  present  them  in  a  dogmatic 
exposition,  by  a  purely  abstract  method,  in  a  single  and 


46  ANALYSIS,   OR  THE   CALCULUS. 

continuous  system.  It  is  this  which  will  be  undertaken 
in  the  present  and  the  five  following  chapters,  limiting  our 
investigations  to  the  most  general  considerations  upon 
each  principal  branch  of  the  science  of  the  calculus. 

The  definite  object  of  our  researches  in  concrete  math- 
ematics being  the  discovery  of  the  equations  which  ex- 
press the  mathematical  laws  of  the  phenomenon  under 
consideration,  and  these  equations  constituting  the  true 
starting  point  of  the  calculus,  which  has  for  its  object 
to  obtain  from  them  the  determination  of  certain  quan- 
tities by  means  of  others,  I  think  it  indispensable,  be 
fore  proceeding  any  farther,  to  go  more  deeply  than  has 
been  customary  into  that  fundamental  idea  of  equation, 
the  continual  subject,  either  as  end  .or  as  beginning,  of 
all  mathematical  labours.  Besides  the  advantage  of  cir- 
cumscribing more  definitely  the  true  field  of  analysis, 
there  will  result  from  it  the  important  consequence  of 
tracing  in  a  more  exact  manner  the  real  line  of  demar- 
cation between  the  concrete  and  the  abstract  part  of 
mathematics,  which  will  complete  the  general  exposition 
of  the  fundamental  division  established  in  the  introduc- 
tory chapter. 

THE  TRUE  IDEA  OF  AN  EQUATION. 

We  usually  form  much  too  vague  an  idea  of  what  an 
equation  is,  when  we  give  that  name  to  every  kind  of 
relation  of  equality  between  any  two  functions  of  the 
magnitudes  which  we  are  considering.  For,  though  ev- 
ery equation  is  evidently  a  relation  of  equality,  it  is  far 
from  being  true  that,  reciprocally,  every  relation  of  equal- 
ity is  a  veritable  equation,  of  the  kind  of  those  to  which, 
by  their  nature,  the  methods  of  analysis  are  applicable. 


THE   TRUE   IDEA   OF   AN   EQUATION.         47 

This  want  of  precision  in  the  logical  consideration  of 
an  idea  which  is  so  fundamental  in  mathematics,  brings 
with  it  the  serious  inconvenience  of  rendering  it  almost 
impossible  to  explain,  in  general  terms,  the  great  and 
fundamental  difficulty  which  we  find  in  establishing  the 
relation  between  the  concrete  and  the  abstract,  and  which 
stands  out  so  prominently  in  each  great  mathematical 
question  taken  by  itself.  If  the  meaning  of  the  word 
equation  was  truly  as  extended  as  we  habitually  suppose 
it  to  be  in  our  definition  of  it,  it  is  not  apparent  what 
great  difficulty  there  could  really  be,  in  general,  in  estab- 
lishing the  equations  of  any  problem  whatsoever  ;  for  the 
whole  would  thus  appear  to  consist  in  a  simple  question 
of  form,  which  ought  never  even  to  exact  any  great  in- 
tellectual efforts,  seeing  that  we  can  hardly  conceive  of 
any  precise  relation  which  is  not  immediately  a. certain 
relation  of  equality,  or  which  cannot  be  readily  brought 
thereto  by  some  very  easy  transformations. 

Thus,  when  we  admit  every  species  of  functions  into 
the  definition  of  equations,  we  do  not  at  all  account  for 
the  extreme  difficulty  which  we  almost  always  experi- 
ence in  putting  a  problem  into  an  equation,  and  which 
so  often  may  be  compared  to  the  efforts  required  by  the 
analytical  elaboration  of  the  equation  when  once  obtain- 
ed. In  a  word,  the  ordinary  abstract  and  general  idea 
of  an  equation  does  not  at  all  correspond  to  the  real 
meaning  which  geometers  attach  to  that  expression  in 
the  actual  development  of  the  science.  Here,  then,  is  a 
logical  fault,  a  defect  of  correlation,  which  it  is  very  im- 
portant to  rectify. 

Division  of  Functions  into  Abstract  and  Concrete. 
To  succeed  in  doing  so,  I  begin  by  distinguishing  two 


48      ANALYSIS,  OR  THE  CALCULUS. 

sorts  of  functions,  abstract  or  analytical  functions,  and 
concrete  functions.  The  first  alone  can  enter  into  ver- 
itable equations.  We  may,  therefore,  henceforth  define 
every  equation,  in  an  exact  and  sufficiently  profound  man- 
ner, as  a  relation  of  equality  between  two  abstract  func- 
tions of  the  magnitudes  under  consideration.  In  order  not 
to  have  to  return  again  to  this  fundamental  definition,  I 
must  add  here,  as  an  indispensable  complement,  without 
which  the  idea  would  not  be  sufficiently  general,  that 
these  abstract  functions  may  refer  not  only  to  the  mag- 
nitudes which  the  problem  presents  of  itself,  but  also  to 
all  the  other  auxiliary  magnitudes  which  are  connected 
with  it,  and  which  we  will  often  be  able  to  introduce, 
simply  as  a  mathematical  artifice,  with  the  sole  object 
of  facilitating  the  discovery  of  the  equations  of  the  phe- 
nomena. I  here  anticipate  summarily  the  result  of  a 
general  discussion  of  the  highest  importance,  which  will 
be  found  at  the  end  of  this  chapter.  We  will  now  re- 
turn to  the  essential  distinction  of  functions  as  abstract 
and  concrete. 

This  distinction  may  be  established  in  two  ways,  es- 
sentially different,  but  complementary  of  each  other,  a 
priori  and  a  posteriori ;  that  is  to  say,  by  characteriz- 
ing in  a  general  manner  the  peculiar  nature  of  each  spe- 
cies of  functions,  and  then  by  making  the  actual  enu- 
meration of  all  the  abstract  functions  at  present  known, 
at  least  so  far  as  relates  to  the  elements  of  which  they 
are  composed. 

A  priori,  the  functions  which  I  call  abstract  are  those 
which  express  a  manner  of  dependence  between  magni- 
tudes, which  can  be  conceived  between  numbers  alone, 
without  there  being  need  of  indicating  any  phenomenon 


ABSTRACT   AND   CONCRETE   FUNCTIONS.  49 

whatever  in  which  it  is  realized.  I  name,  on  the  other 
hand,  concrete  functions,  those  for  which  the  mode  of  de- 
pendence expressed  cannot  be  defined  or  conceived  except 
by  assigning  a  determinate  case  of  physics,  geometry,  me- 
chanics, &c.,  in  which  it  actually  exists. 

Most  functions  in  their  origin,  even  those  which  are 
at  present  the  most  purely  abstract,  have  begun  by  be- 
ing concrete  ;  so  that  it  is  easy  to  make  the  preceding 
distinction  understood,  by  citing  only  the  successive  dif- 
ferent points  of  view  under  which,  in  proportion  as  the 
science  has  become  formed,  geometers  have  considered 
the  most  simple  analytical  functions.  I  will  indicate 
powers,  for  example,  which  have  in  general  become  ab- 
stract functions  only  since  the  labours  of  Vieta  and  Des- 
cartes. The  functions  x2,  x3,  which  in  our  present  anal- 
ysis are  so  well  conceived  as  simply  abstract,  were,  for 
the  geometers  of  antiquity,  perfectly  concrete  functions, 
expressing  the  relation  of  the  superficies  of  a  square,  or 
the  volume  of  a  cube  to  the  length  of  their  side.  These 
had  in  their  eyes  such  a  character  so  exclusively,  that 
it  was  only  by  means  of  the  geometrical  definitions  that 
they  discovered  the  elementary  algebraic  properties  of 
these  functions,  relating  to  the  decomposition  of  the 
variable  into  two  parts,  properties  which  were  at  that 
epoch  only  real  theorems  of  geometry,  to  which  a  nu- 
merical meaning  was  not  attached  until  long  after- 
ward. 

I  shall  have  occasion  to  cite  presently,  for  another  rea- 
son, a  new  example,  very  suitable  to  make  apparent  the 
fundamental  distinction  which  I  have  just  exhibited  ;  it 
is  that  of  circular  functions,  both  direct  and  inverse,  which 
at  the  present  time  are  still  sometimes  concrete,  some- 

D 


50      ANALYSIS,  OR  THE  CALCULUS. 

times  abstract,  according  to  the  point  of  view  under  which 
they  are  regarded. 

A  posteriori,  the  general  character  which  renders  a 
function  abstract  or  concrete  having  been  established,  the 
question  as  to  whether  a  certain  determinate  function  is 
veritably  abstract,  and  therefore  susceptible  of  entering 
:nto  true  analytical  equations,  becomes  a  simple  question 
->f  fact,  inasmuch  as  we  are  going  to  enumerate  all  the 
functions  of  this  species. 

Enumeration  of  Abstract  Functions.  At  first  view 
this  enumeration  seems  impossible,  the  distinct  analyt- 
ical functions  being  infinite  in  number.  But  when  we 
divide  them  into  simple  and  compound,  the  difficulty  dis- 
appears ;  for,  though  the  number  of  the  different  func- 
tions considered  in  mathematical  analysis  is  really  infi- 
nite,  they  are,  on  the  contrary,  even  at  the  present  day, 
composed  of  a  very  small  number  of  elementary  functions, 
which  can  be  easily  assigned,  and  which  are  evidently 
ufficient  for  deciding  the  abstract  or  concrete  character 
>f  any  given  function ;  which  will  be  of  the  one  or  the 
other  nature,  according  as  it  shall  be  composed  exclusive- 
ly of  these  simple  abstract  functions,  or  as  it  shall  in- 
clude others. 

We  evidently  have  to  consider,  for  this  purpose,  only 
.he  functions  of  a  single  variable,  since  those  relative 
to  several  independent  variables  are  constantly,  by  their 
nature,  more  or  less  compound. 

Let  x  be  the  independent  variable,  y  the  correlative 
variable  which  depends  upon  it.  The  different  simple 
modes  of  abstract  dependence,  which  we  can  now  conceive 
between  y  and  x,  are  expressed  by  the  ten  following  el- 
ementary formulas,  in  which  each  function  is  coupled 


ENUMERATION   OF    FUNCTIONS.  5} 

with  its  inverse,  that  is,  with  that  which  would  be  ob- 
tained from  the  direct  function  by  referring  x  to  y,  in- 
stead of  referring  y  to  x. 

FUNCTION.  ITS  NAME 

(  1°  y=a+x Sum. 

1st  couple  <  _0 

(  2    y=a—x Difference. 

(  1°  y=ax Product. 

2d  couple    <  Oo        a  r\     4- 

)  2    y=- Quotient. 

(  x 

(  1°  y=x" Power. 

3d  couple    <  _ 

1  2°  y=Vx Root. 

(  1°  y=a? Exponential. 

4th  couple  <  rtQ 

(  2    y=lx Logarithmic. 

(  1°  v=sin.  x Direct  Circular. 

oth  couple  < 

(  2    #=arc(sm.—  x\  .  Inverse  Circular.* 

Such  are  the  elements,  very  few  in  number,  which  di- 
rectly compose  all  the  abstract  functions  known  at  the 
present  day.  Few  as  they  are,  they  are  evidently  suf- 
ficient to  give  rise  to  an  infinite  number  of  analytical 
combinations. 


*  With  the  view  of  increasing  as  much  as  possible  the  resources  and  the 
extent  (now  so  insufficient)  of  mathematical  analysis,  geometers  count  this 
iast  couple  of  functions  among  the  analytical  elements.  Although  this  in- 
scription is  strictly  legitimate,  it  is  important  to  remark  that  circular  func- 
tions are  not  exactly  in  the  same  situation  as  the  other  abstract  elementary 
functions.  There  is  this  very  essential  difference,  that  the  functions  of  the 
four  first  couples  are  at  the  same  time  simple  and  abstract,  while  the  circu- 
lar functions,  which  may  manifest  each  character  in  succession,  according 
to  the  point  of  view  under  which  they  are  considered  and  the  manner  in 
which  they  are  employed,  never  present  these  two  properties  simultane- 
ously. 

Some  other  concrete  functions  may  be  usefully  introduced  into  the  num- 
ber of  analytical  elements,  certain  conditions  being  fulfilled.  It  is  thus,  for 
example,  that  the  labours  of  M.  Legendre  and  of  M.  Jacobi  on  elliptical 
functions  have  truly  enlarged  the  field  of  analysis ;  and  the  same  is  true  of 
some  definite  integrals  obtained  by  M.  Fourier  in  the  theory  of  heat. 


52      ANALYSIS,  OR  THE  CALCULUS. 

No  rational  consideration  rigorously  circumscribes,  a 
priori,  the  preceding  table,  which  is  only  the  actual  ex- 
pression of  the  present  state  of  the  science.  Our  ana- 
lytical elements  are  at  the  present  day  more  numerous 
than  they  were  for  Descartes,  and  even  for  Newton  and 
Leibnitz  :  it  is  only  a  century  since  the  last  two  couples 
have  been  introduced  into  analysis  by  the  labours  of  John 
Bernouilli  and  Euler.  Doubtless  new  ones  will  be  here- 
after admitted  ;  but,  as  I  shall  show  towards  the  end  of 
this  chapter,  we  cannot  hope  that  they  will  ever  be  great- 
ly  multiplied,  their  real  augmentation  giving  rise  to  very 
great  difficulties. 

We  can  now  form  a  definite,  and,  at  the  same  time, 
sufficiently  extended  idea  of  what  geometers  understand 
by  a  veritable  equation.  This  explanation  is  especially 
suited  to  make  us  understand  how  difficult  it  must  be 
really  to  establish  the  equations  of  phenomena,  since, we 
have  effectually  succeeded  in  so  doing  only  when  we 
have  been  able  to  conceive  the  mathematical  laws  of 
these  phenomena  by  the  aid  of  functions  entirely  com- 
posed of  only  the  mathematical  elements  which  I  have 
just  enumerated.  It  is  clear,  in  fact,  that  it  is  then 
only  that  the  problem  becomes  truly  abstract,  and  is  re- 
duced to  a  pure  question  of  numbers,  these  functions 
being  the  only  simple  relations  which  we  can  conceive 
between  numbers,  considered  by  themselves.  Up  to  this 
period  of  the  solution,  whatever  the  appearances  may  be, 
the  question  is  still  essentially  concrete,  and  does  not  come 
within  the>  domain  of  the  calculus.  Now  the  fundamen- 
tal difficulty  of  this  passage  from  the  concrete  to  the  ab- 
stract in  general  consists  especially  in  the  insufficiency 
of  this  very  small  number  of  analytical  elements  which 


ITS   TWO    PRINCIPAL   DIVISIONS.  53 

we  possess,  and  by  means  of  which,  nevertheless,  in  spite 
of  the  little  real  variety  which  they  offer  us,  we  must 
succeed  in  representing  all  the  precise  relations  which 
all  the  different  natural  phenomena  can  manifest  to  us. 
Considering  the  infinite  diversity  which  must  necessa- 
rily exist  in  this  respect  in  the  external  world,  we  easily 
understand  how  far  below  the  true  difficulty  our  con- 
ceptions must  frequently  be  found,  especially  if  we  add 
that  as  these  elements  of  our  analysis  have  been  in  the 
first  place  furnished  to  us  by  the  mathematical  consid- 
eration of  the  simplest  phenomena,  we  have,  a  priori,  no 
rational  guarantee  of  their  necessary  suitableness  to  rep- 
resent the  mathematical  law  of.  every  other  class  of  phe- 
nomena. I  will  explain  presently  the  general  artifice,  so 
profoundly  ingenious,  by  which  the  human  mind  has  suc- 
ceeded in  diminishing,  in  a  remarkable  degree,  this  fun- 
damental difficulty  which  is  presented  by  the  relation  of 
the  concrete  to  the  abstract  in  mathematics,  without, 
however,  its  having  been  necessary  to  multiply  the  num- 
ber of  these  analytical  elements.  * 

THE    TWO   PRINCIPAL   DIVISIONS   OF    THE    CALCULUS. 

The  preceding  explanations  determine  with  precision 
the  true  object  and  the  real  field  of  abstract  mathemat- 
ics. I  must  now  pass  to  the  examination  of  its  princi- 
pal divisions,  for  thus  far  we  have  considered  the  calcu- 
lus as  a  whole. 

The  first  direct  consideration  to  be  presented  on  the 
composition  of  the  science  of  the  calculus  consists  in  di- 
viding it,  in  the  first  place,  into  two  principal  branches, 
to  which,  for  want  of  more  suitable  denominations,  I  will 
give  the  names  of  Algebraic  calculus,  or  Algebra,  and  of 


54      ANALYSIS,  OR  THE  CALCULUS. 

Arithmetical  calculus,  or  Arithmetic  ;  but  with  the  cau- 
tion to  take  these  two  expressions  in  their  most  extended 
logical  acceptation,  in  the  place  of  the  by  far  too  restrict- 
ed meaning  which  is  usually  attached  to  them. 

The  complete  solution  of  every  question  of  the  calcu- 
lus, from  the  most  elementary  up  to  the  most  transcend- 
ental, is  necessarily  composed  of  two  successive  parts, 
whose  nature  is  essentially  distinct.  In  the  first,  the  ob- 
ject is  to  transform  the  proposed  equations,  so  as  to  make 
apparent  the  manner  in  which  the  unknown  quantities 
are  formed  by  the  known  ones :  it  is  this  which  consti- 
tutes the  algebraic  question.  In  the  second,  our  object 
is  to  find  the  values  of  the  formulas  thus  obtained  ;  that 
is,  to  determine  directly  the  values  of  the  numbers  sought, 
which  are  already  represented  by  certain  explicit  func- 
tions of  given  numbers  :  this  is  the  arithmetical  ques- 
tion.*1 It  is  apparent  that,  in  every  solution  which  is 

*  Suppose,  for  example,  that  a  question  gives  the  following  equation  be- 
tween an  unknown  magnitude  x,  and  two  known  magnitudes,  a  and  b, 

xs+3ax=2b, 

as  is  the  case  in  the  problem  of  the  trisection  of  an  angle.  We  see  at  once 
that  the  dependence  between  x  on  the  one  side,  and  db  on  the  other,  is 
completely  determined ;  but,  so  long  as  the  equation  preserves  its  primitive 
form,  we  do  not  at  all  perceive  in  what  manner  the  unknown  quantity  is 
derived  from  the  data.  This  must  be  discovered,  however,  before  we  can 
think  of  determining  its  value.  Such  is  the  object  of  the  algebraic  part  of 
the  solution.  When,  by  a  series  of  transformations  which  have  successively 
rendered  that  derivation  more  and  more  apparent,  we  have  arrived  at  pre- 
senting the  proposed  equation  under  the  form 


the  work  of  algebra  is  finished ;  and  even  if  we  could  not  perform  the  arith- 
metical operations  indicated  by  that  formula,  we  would  nevertheless  have 
obtained  a  knowledge  very  real,  and  often  very  important.  The  work  of 
arithmetic  will  now  consist  in  taking  that  formula  for  its  starting  point,  and 
finding  the  number  x  when  the  values  of  the  numbers  a  and  b  are  given. 


ITS   TWO    PRINCIPAL   DIVISIONS.  55 

truly  rational,  it  necessarily  follows  the  algebraical  ques- 
tion, of  which  it  forms  the  indispensable  complement, 
since  it  is  evidently  necessary  to  know  the  mode  of  gener- 
ation of  the  numbers  sought  for  before  determining  their 
actual  values  for  each  particular  case.  Thus  the  stop- 
ping-place of  the  algebraic  part  of  the  solution  becomes 
the  starting  point  of  the  arithmetical  part. 

We  thus  see  that  the  algebraic  calculus  and  the  arith- 
metical calculus  differ  essentially  in  their  object.  They 
differ  no  less  in  the  point  of  view  under  which  they  regard 
quantities ;  which  are  considered  in  the  first  as  to  their 
relations,  and  in  the  second  as  to  their  values.  The 
true  spirit  of  the  calculus,  in  general,  requires  this  dis- 
tinction to  be  maintained  with  the  most  severe  exacti- 
tude, and  the  line  of  demarcation  between  the  two  peri- 
ods of  the  solution  to  be  rendered  as  clear  and  distinct 
as  the  proposed  question  permits.  The  attentive  obser- 
vation of  this  precept,  which  is  too  much  neglected,  may 
be  of  much  assistance,  in  each  particular  question,  in  di- 
recting the  efforts  of  our  mind,  at  any  moment  of  the 
solution,  towards  the  real  corresponding  difficulty.  In 
truth,  the  imperfection  of  the  science  of  the  calculus 
obliges  us  very  often  (as  will  be  explained  in  the  next 
chapter)  to  intermingle  algebraic  and  arithmetical  consid- 
erations in  the  solution  of  the  same  question.  But,  how- 
ever impossible  it  may  be  to  separate  clearly  the  two  parts 
of  the  labour,  yet  the  preceding  indications  will  always 
enable  us  to  avoid  confounding  them. 

In  endeavouring  to  sum  up  as  succinctly  as  possible 
the  distinction  just  established,  we  see  that  ALGEBRA 
may  be  defined,  in  general,  as  having  for  its  object  the 
resolution  of  equations ;  taking  this  expression  in  its 


56  ANALYSIS,   OR  THE  CALCULUS. 

full  logical  meaning,  which  signifies  the  transformation 
of  implicit  functions  into  equivalent  explicit  ones.  In 
the  same  way,  ARITHMETIC  may  be  denned  as  destined 
to  the  determination  of  the  values  of  functions.  Hence- 
forth, therefore,  we  will  briefly  say  that  ALGEBRA  is  the 
Calculus  of  Functions,  and  ARITHMETIC  the  Calculus  of 
Values. 

We  can  now  perceive  how  insufficient  and  even  erro- 
neous are  the  ordinary  definitions.  Most  generally,  the 
exaggerated  importance  attributed  to  Signs  has  led  to  the 
distinguishing  the  two  fundamental  branches  of  the  sci- 
ence of  the  Calculus  by  the  manner  of  designating  in 
each  the  subjects  of  discussion,  an  idea  which  is  evident- 
ly absurd  in  principle  and  false  in  fact.  Even  the  cele- 
brated definition  given  by  Newton,  characterizing  Alge- 
bra as  Universal  Arithmetic,  gives  certainly  a  very  false 
idea  of  the  nature  of  algebra  and  of  that  of  arithmetic.^ 

Having  thus  established  the  fundamental  division  of 
the  calculus  into  two  principal  branches,  I  have  now  to 
compare  in  general  terms  the  extent,  the  importance,  and 
the  difficulty  of  these  two  sorts  of  calculus,  so  as  to  have 
hereafter  to  consider  only  the  Calculus  of  Functions, 
which  is  to  be  the  principal  subject  of  our  study. 


*  I  have  thought  that  I  ought  to  specially  notice  this  definition,  because 
it  serves  as  the  basis  of  the  opinion  which  many  intelligent  persons,  unac 
quainted  with  mathematical  science,  form  of  its  abstract  part,  without  con 
sidering  that  at  the  time  of  this  definition  mathematical  analysis  was  not 
sufficiently  developed  to  Enable  the  general  character  of  each  of  its  princi- 
pal parts  to  be  properly  apprehended,  which  explains  why  Newton  could 
at  that  time  propose  a  definition  which  at  the  present  day  he  would  cer- 
tainly reject. 


THE  CALCULUS  OF  VALUES.  57 

THE    CALCULUS   OF   VALUES,    OR   ARITHMETIC. 

Its  Extent.  The  Calculus  of  Values,  or  Arithmetic, 
would  appear,  at  first  view,  to  present  a  field  as  vast  as 
that  of  algebra,  since  it  would  seem  to  admit  as  many 
distinct  questions  as  we  can  conceive  different  algebraic 
formulas  whose  values  are  to  be  determined.  But  a  very 
simple  reflection  will  show  the  difference.  Dividing  func- 
tions into  simple  and  compound,  it  is  evident  that  when 
we  know  how  to  determine  the  value  of  simple  functions, 
the  consideration  of  compound  functions  will  no  longer 
present  any  difficulty.  In  the  algebraic  point  of  view, 
a  compound  function  plays  a  very  different  part  from  that 
of  the  elementary  functions  of  which  it  consists,  and  from 
this,  indeed,  proceed  all  the  principal  difficulties  of  analy- 
sis. But  it  is  very  different  with  the  Arithmetical  Cal- 
culus. Thus  the  number  of  truly  distinct  arithmetical 
operations  is  only  that  determined  by  the  number  of  the 
elementary  abstract  functions,  the  very  limited  list  of 
which  has  been  given  above.  The  determination  of  the 
values  of  these  ten  functions  necessarily  gives  that  of  all 
the  functions,  infinite  in  number,  which  are  considered 
in  the  whole  of  mathematical  analysis,  such  at  least  as 
it  exists  at  present.  There  can  be  no  new  arithmetical 
operations  without  the  creation  of  really  new  analytical 
elements,  the  number  of  which  must  always  be  extreme- 
ly small.  The  field  of  arithmetic  is,  then,  by  its  nature, 
exceedingly  restricted,  while  that  of  algebra  is  rigorously 
indefinite. 

It  is,  however,  important  to  remark,  that  the  domain 
of  the  calculus  of  values  is,  in  reality,  much  more  ex- 
tensive than  it  is  commonly  represented  ;  for  several  ques- 


58      ANALYSIS,  OR  THE  CALCULUS. 

tions  truly  arithmetical,  since  they  consist  of  determi- 
nations of  values,  are  not  ordinarily  classed  as  such,  be- 
cause we  are  accustomed  to  treat  them  only  as  inci- 
dental in  the  midst  of  a  body  of  analytical  researches 
more  or  less  elevated,  the  too  high  opinion  commonly 
formed  of  the  influence  of  signs  being  again  the  princi- 
pal cause  of  this  confusion  of  ideas.  Thus  not  only  the 
construction  of  a  table  of  logarithms,  but  also  the  calcu- 
lation of  trigonometrical  tables,  are  true  arithmetical  op- 
erations of  a  higher  kind.  We  may  also  cite  as  being 
in  the  same  class,  although  in  a  very  distinct  and  more 
elevated  order,  all  the  methods  by  which  we  determine 
directly  the  value  of  any  function  for  each  particular  sys- 
tem of  values  attributed  to  the  quantities  on  which  it  de- 
pends, when  we  cannot  express  in  general  terms  the  ex- 
plicit form  of  that  function.  In  this  point  of  view  the 
numerical  solution  of  questions  which  we  cannot  resolve 
algebraically,  and  even  the  calculation  of  "  Definite  In- 
tegrals," whose  general  integrals  we  do  not  know,  really 
make  a  part,  in  spite  of  all  appearances,  of  the  domain 
of  arithmetic,  in  which  we  must  necessarily  comprise  all 
that  which  has  for  its  object  the  determination  of  the 
values  of  functions.  The  considerations  relative  to  this 
object  are,  in  fact,  constantly  homogeneous,  whatever  the 
determinations  in  question,  and  are  always  very  distinct 
from  truly  algebraic  considerations. 

To  complete  a  just  idea  of  the  real  extent  of  the  cal- 
culus of  values,  we  must  include  in  it  likewise  that  part 
of  the  general  science  of  the  calculus  which  now  bears 
the  name  of  the  Theory  of  Numbers,  and  which  is  yet 
so  little  advanced.  This  branch,  very  extensive  by  its 
nature,  but  whose  importance  in  the  general  system  of 


THE  CALCULUS  OF  VALUES.       59 

science  is  not  very  great,  has  for  its  object  the  discovery 
of  the  properties  inherent  in  different  numbers  by  virtue 
of  their  values,  and  independent  of  any  particular  sys- 
tem of  numeration.  It  forms,  then,  a  sort  of  transcen- 
dental arithmetic  ;  and  to  it  would  really  apply  the  def- 
inition proposed  by  Newton  for  algebra. 

The  entire  domain  of  arithmetic  is,  then,  much  more 
extended  than  is  commonly  supposed  ;  but  this  calculus 
of  values  will  still  never  be  more  than  a  point,  so  to 
speak,  in  comparison  with  the  calculus  of  functions,  of 
which  mathematical  science  essentially  consists.  This 
comparative  estimate  will  be  still  more  apparent  from 
some  considerations  which  I  have  now  to  indicate  re- 
specting the  true  nature  of  arithmetical  questions  in  gen- 
eral, when  they  are  more  profoundly  examined. 

Its  true  Nature.  In  seeking  to  determine  with  pre- 
cision in  what  determinations  of  values  properly  consist, 
we  easily  recognize  that  they  are  nothing  else  but  veri- 
table transformations  of  the  functions  to  be  valued  ; 
transformations  which,  in  spite  of  their  special  end,  are 
none  the  less  essentially  of  the  same  nature  as  all  those 
taught  by  analysis.  In  this  point  of  view,  the  calculus 
of  values  might  be  simply  conceived  as  an  appendix,  and 
a  particular  application  of  the  calculus  of  functions,  so 
that  arithmetic  would  disappear,  so  to  say,  as  a  distinct 
section  in  the  whole  body  of  abstract  mathematics. 

In  order  thoroughly  to  comprehend  this  consideration, 
we  must  observe  that,  when  we  propose  to  determine  the 
value  of  an  unknown  number  whose  mode  of  formation  is 
given,  it  is,  by  the  mere  enunciation  of  the  arithmetical 
question,  already  defined  and  expressed  under  a  certain 
form  ;  and  that  in  determining  its  vatue  we  only  put  its 


60      ANALYSIS,  OR  THE  CALCULUS. 

expression  under  another  determinate  form,  to  whicn  tro 
are  accustomed  to  refer  the  exact  notion  of  each  particu- 
lar number  by  making  it  re-enter  into  the  regular  system 
of  numeration.  The  determination  of  values  consists 
so  completely  of  a  simple  transformation,  that  when  the 
primitive  expression  of  the  number  is  found  to  be  already 
conformed  to  the  regular  system  of  numeration,  there 
is  no  longer  any  determination  of  value,  properly  speak- 
ing, or,  rather,  the  question  is  answered  by  the  question 
itself.  Let  the  question  be  to  add  the  two  numbers  one 
and  twenty,  we  answer  it  by  merely  repeating  the  enun- 
ciation of  the  question,^  and  nevertheless  we  think  that 
we  have  determined  the  value  of  the  sum.  This  signi- 
fies that  in  this  case  the  first  expression  of  the  function 
had  no  need  of  being  transformed,  while  it  would  not  be 
thus  in  adding  twenty-three  and  fourteen,  for  then  the 
sum  would  not  be  immediately  expressed  in  a  manner 
conformed  to  the  rank  which  it  occupies  in  the  fixed  and 
general  scale  of  numeration. 

To  sum  up  as  comprehensively  as  possible  the  preced- 
ing views,  we  may  say,  that  to  determine  the  value  of 
a  number  is  nothing  else  than  putting  its  primitive  ex- 
pression under  the  form 

a+bz+czz+dz3-\-ez* +pzm, 

z  being  generally  equal  to  10,  and  the  coefficients  <z,  b, 
c,  d,  &c.,  being  subjected  to  the  conditions  of  being  whole 
numbers  less  than  z  ;  capable  of  becoming  equal  to  zero  ; 
but  never  negative.  Every  arithmetical  question  may 
thus  be  stated  as  consisting  in  putting  under  such  a  form 

*  This  is  less  strictly  true  in  the  English  system  of  numeration  than  in 
the  French,  since  "  twenty-one"  is  our  more  usual  mode  of  expressing  this 
number. 


THE  CALCULUS  OF  FUNCTIONS.      g] 

any  abstract  function  whatever  of  different  quantities, 
which  are  supposed  to  have  themselves  a  similar  form 
already.  We  might  then  see  in  the  different  operations 
of  arithmetic  only  simple  particular  cases  of  certain  alge- 
braic transformations,  excepting  the  special  difficulties 
belonging  to  conditions  relating  to  the  nature  of  the  co- 
efficients. 

It  clearly  follows  that  abstract  mathematics  is  essen- 
tially composed  of  the  Calculus  of  Functions,  which  had 
been  already  seen  to  be  its  most  important,  most  extend- 
ed, and  most  difficult  part.  It  will  henceforth  be  the  ex- 
clusive subject  of  our  analytical  investigations.  I  will 
therefore  no  longer  delay  on  the  Calculus  of  Values,  but 
pass  immediately  to  the  examination  of  the  fundamental 
division  of  the  Calculus  of  Functions. 

THE    CALCULUS   OF    FUNCTIONS,   OR   ALGEBRA. 


Principle  of  its  Fundamental  Division.  We  have 
determined,  at  the  beginning  of  this  chapter,  wherein 
properly  consists  the  difficulty  which  we  experience  in 
putting  mathematical  questions  into  equations.  It  is  es- 
sentially because  of  the  insufficiency  of  the  very  small 
number  of  analytical  elements  which  we  possess,  that 
the  relation  of  the  concrete  to  the  abstract  is  usually  so 
difficult  to  establish.  Let  us  endeavour  now  to  appre- 
ciate in  a  philosophical  manner  the  general  process  by 
which  the  human  mind  has  succeeded,  in  so  great  a  num- 
ber of  important  cases,  in  overcoming  this  fundamental 
obstacle  to  The  establishment  of  Equations. 

1.  By  the  Creation  of  new  Functions.  In  looking  at 
this  important  question  from  the  most  general  point  of 
view,  we  are  led  at  once  to  the  conception  of  one  means  of 


62      ANALYSIS,  OR  THE  CALCULUS. 

facilitating  the  establishment  of  the  equations  of  phenom- 
ena. Since  the  principal  obstacle  in  this  matter  comes 
from  the  too  small  number  of  our  analytical  elements,  the 
whole  question  would  seem  to  be  reduced  to  creating 
new  ones.  But  this  means,  though  natural,  is  really 
illusory ;  and  though  it  might  be  useful,  it  is  certainly 
insufficient. 

In  fact,  the  creation  of  an  elementary  abstract  func- 
tion, which  shall  be  veritably  new,  presents  in  itself  the 
greatest  difficulties.  There  is  even  something  contra- 
dictory in  such  an  idea ;  for  a  new  analytical  element 
would  evidently  not  fulfil  its  essential  and  appropriate 
conditions,  if  we  could  not  immediately  determine  its 
value.  Now,  on  the  other  hand,  how  are  we  to  deter- 
mine the  value  of  a  new  function  which  is  truly  simple, 
that  is,  which  is'not  formed  by  a  combination  of  those 
already  known  ?  That  appears  almost  impossible.  The 
introduction  into  analysis  of  another  elementary  abstract 
function,  or  rather  of  another  couple  of  functions  (for  each 
would  be  always  accompanied  by  its  inverse),  supposes 
then,  of  necessity,  the  simultaneous  creation  of  a  new 
arithmetical  operation,  which  is  certainly  very  difficult. 

If  we  endeavour  to  obtain  an  idea  of  the  means  which 
the  human  mind  employs  for  inventing  new  analytical 
elements,  by  the  examination  of  the  procedures  by  the 
aid  of  which  it  has  actually  conceived  those  which  we 
already  possess,  our  observations  leave  us  in  that  respect 
in  an  entire  uncertainty,  for  the  artifices  which  it  has 
already  made  use  of  for  that  purpose  are  evidently  ex- 
hausted. To  convince  ourselves  of  it,  let  us  consider 
the  last  couple  of  simple  functions  which  has  been  in- 
troduced into  analysis,  and  at  the  formation  of  which  we 


THE  CALCULUS  OF  FUNCTIONS.      53 

have  been  present,  so  to  speak,  namely,  the  fourth  couple  ; 
for,  as  I  have  explained,  the  fifth  couple  does  not  strictly 
give  veritable  new  analytical  elements.  The  function 
ax,  and,  consequently,  its  inverse,  have  been  formed  by 
conceiving,  under  a  new  point  of  view,  a  function  which 
had  been  a  long  time  known,  namely,  powers — when  the 
idea  of  them  had  become  sufficiently  generalized.  The 
consideration  of  a  power  relatively  to  the  variation  of  its 
exponent,  instead  of  to  the  variation  of  its  base,  was  suf- 
ficient to  give  rise  to  a  truly  novel  simple  function,  the 
variation  following  then  an  entirely  different  route.  But 
this  artifice,  as  simple  as  ingenious,  can  furnish  nothing 
more  ;  for,  in  turning  over  in  the  same  manner  all  our 
present  analytical  elements,  we  end  in  only  making  them 
return  into  one  another. 

"We  have,  then,  no  idea  as  to  how  we  could  proceed  to 
the  creation  of  new  elementary  abstract  functions  which 
would  properly  satisfy  all  the  necessary  conditions.  This 
is  not  to  say,  however,  that  we  have  at  present  attain, 
ed  the  effectual  limit  established  in  that  respect  by  the 
bounds  of  our  intelligence.  It  is  even  certain  that  the 
last  special  improvements  in  mathematical  analysis  have 
contributed  to  extend  our  resources  in  that  respect,  by 
introducing  within  the  domain  of  the  calculus  certain  def- 
inite integrals,  which  in  some  respects  supply  the  place 
of  new  simple  functions,  although  they  are  far  from  ful- 
filling all  the  necessary  conditions,  which  has  prevented 
me  from  inserting  them  in  the  table  of  true  analytical 
elements.  But,  on  the  whole,  I  think  it  unquestionable 
that  the  number  of  these  elements  cannot  increase  ex- 
cept with  extreme  slowness.  It  is  therefore  not  from 
these  sources  that  the  human  mind  has  drawn  its  most 


64  ANALYSIS,   OR  THE   CALCULUS. 

powerful  means  of  facilitating,  as  much  as  is  possible, 
the  establishment  of  equations. 

2.  By  the  Conception  of  Equations  between  certain 
auxiliary  Quantities.  This  first  method  being  set  aside, 
there  remains  evidently  but  one  other :  it  is,  seeing  the 
impossibility  of  finding  directly  the  equations  between 
the  quantities  under  consideration,  to  seek  for  correspond- 
ing ones  between  other  auxiliary  quantities,  connected 
with  the  first  according  to  a  certain  determinate  law, 
and  from  the  relation  between  which  we  may  return  to 
that  between  the  primitive  magnitudes.  Such  is,  in 
substance,  the  eminently  fruitful  conception  which  the 
human  mind  has  succeeded  in  establishing,  and  which 
constitutes  its  most  admirable  instrument  for  the  mathe- 
matical explanation  of  natural  phenomena ;  the  analysis, 
called  transcendental. 

As  a  general  philosophical  principle,  the  auxiliary 
quantities,  which  are  introduced  in  the  place  of  the  prim- 
itive magnitudes,  or  concurrently  with  them,  in  order  to 
facilitate  the  establishment  of  equations,  might  be  de- 
rived according  to  any  law  whatever  from  the  immediate 
elements  of  the  question.  This  conception  has  thus  a 
much  more  extensive  reach  than  has  been  commonly  at- 
tributed to  it  by  even  the  most  profound  geometers.  It 
is  extremely  important  for  us  to  view  it  in  its  whole  log- 
ical extent,  for  it  will  perhaps  be  by  establishing  a  gen- 
eral mode  of  derivation  different  from  that  to  which  we 
have  thus  far  confined  ourselves  (although  it  is  evidently 
very  far  from  being  the  only  possible  one)  that  we  shall 
one  day  succeed  in  essentially  perfecting  mathematical 
analysis  as  a  whole,  and  consequently  in  establishing 
more  powerful  means  of  investigating  the  laws  of  nature 


THE  CALCULUS  OF  FUNCTIONS.     55 

than  our  present  processes,  which  are  unquestionably  sus- 
ceptible of  becoming  exhausted. 

But,  regarding  merely  the  present  constitution  of  the 
science,  the  only  auxiliary  quantities  habitually  intro- 
duced in  the  place  of  the  primitive  quantities  in  the 
Transcendental  Analysis  are  what  are  called,  1°,  infi- 
nitely small  elements,  the  differentials  (of  different  or- 
ders) of  those  quantities,  if  we  regard  this  analysis  in  the 
manner  of  LEIBNITZ  ;  or,  2°,  the  fluxions,  the  limits  of 
the  ratios  of  the  simultaneous  increments  of  the  primi- 
tive quantities  compared  with  one  another,  or,  more 
briefly,  the  prime  and  ultimate  ratios  of  these  incre- 
ments, if  we  adopt  the  conception  of  NEWTON  ;  or,  3°, 
the  derivatives,  properly  so  called,  of  those  quantities, 

that  is,  the  coefficients  of  the  different  terms  of  their  re- 

• 

spective  increments,  according  to  the  conception  of  LA- 
GRANGE. 

These  three  principal  methods  of  viewing  our  present 
transcendental  analysis,  and  all  the  other  less  distinctly 
characterized  ones  which  have  been  successively  pro- 
posed, are,  by  their  nature,  necessarily  identical,  whether 
in  the  calculation  or  in  the  application,  as  will  be  ex- 
plained in  a  general  manner  in  the  third  chapter.  As  to 
their  relative  value,  we  shall  there  see  that  the  concep- 
tion of  Leibnitz  has  thus  far,  in  practice,  an  incontesta- 
ble superiority,  but  that  its  logical  character  is  exceed- 
ingly vicious ;  while  that  the  conception  of  Lagrange, 
admirable  by  its  simplicity,  by  its  logical  perfection,  by 
the  philosophical  unity  which  it  has  established  in  math- 
ematical analysis  (till  then  separated  into  two  almost  en- 
tirely independent  worlds),  presents,  as  yet,  serious  incon- 
veniences in  the  applications,  by  retarding  the  progress 

E 


66      ANALYSIS,  OR  THE  CALCULUS. 

of  the  mind.  The  conception  of  Newton  occupies  nearly 
middle  ground  in  these  various  relations,  being  less  rapid, 
but  more  rational  than  that  of  Leibnitz  ;  less  philosoph- 
ical, but  more  applicable  than  that  of  Lagrange. 

This  is  not  the  place  to  explain  the  advantages  of  the 
introduction  of  this  kind  of  auxiliary  quantities  in  the 
place  of  the  primitive  magnitudes.  The  third  chapter 
is  devoted  to  this  subject.  At  present  I  limit  myself  to 
consider  this  conception  in  the  most  general  manner,  in 
order  to  deduce  therefrom  the  fundamental  division  of 
the  calculus  of  functions  into  two  systems  essentially 
distinct,  whose  dependence,  for  the  complete  solution  of 
any  one  mathematical  question,  is  invariably  determi- 
nate. 

In  this  connexion,  and  in  the  logical  order  of  ideas, 
the  transcendental  analysis  presents  itself  as  being  ne- 
cessarily the  first,  since  its  general  object  is  to  facilitate 
the  establishment  of  equations,  an  operation  which  must 
evidently  precede  the  resolution  of  those  equations,  which 
is  the  object  of  the  ordinary  analysis.  But  though  it  is 
exceedingly  important  to  conceive  in  this  way  the  true 
relations  of  these  two  systems  of  analysis,  it  is  none  the 
less  proper,  in  conformity  with  the  regular  usage,  to 
study  the  transcendental  analysis  after  ordinary  analy- 
sis ;  for  though  the  former  is,  at  bottom,  by  itself  log- 
ically independent  of  the  latter,  or,  at  least,  may  be  es- 
sentially disengaged  from  it,  yet  it  is  clear  that,  since  * 
its  employment  in  the  solution  of  questions  has  always 
more  or  less  need  of  being  completed  by  the  use  of  the 
ordinary  analysis,  we  would  be  constrained  to  leave  the 
questions  in  suspense  if  this  latter  had  not  beenvprevious- 
ly  studied. 


THE  CALCULUS  OF  FUNCTIONS.      67 

Corresponding  Divisions  of  the  Calculus  of  Func- 
tions. It  follows  from  the  preceding  considerations  that 
the  Calculus  of  Functions,  or  Algebra  (taking  this  word 
in  its  most  extended  meaning),  is  composed  of  two  dis- 
tinct fundamental  branches,  one  of  which  has  for  its  im- 
mediate object  the  resolution  of  equations,  when  they 
are  directly  established  between  the  magnitudes  them- 
selves which  are  under  consideration ;  and  the  other, 
starting  from  equations  (generally  much  easier  to  form) 
between  quantities  indirectly  connected  with  those  of 
the  problem,  has  for  its  peculiar  and  constant  destina- 
tion the  deduction,  by  invariable  analytical  methods,  of 
the  corresponding  equations  between  the  direct  magni- 
tudes which  we  are  considering  ;  which  brings  the  ques- 
tion within  the  domain  of  the  preceding  calculus. 

The  former  calculus  bears  most  frequently  the  name 
of  Ordinary  Analysis,  or  of  Algebra,  properly  so  called. 
The  second  constitutes  what  is  called  the  Transcendent- 
al Analysis,  which  has  been  designated  by  the  different 
denominations  of  Infinitesimal  Calculus,  Calculus  of 
Fluxions  and  of  Fluents,  Calculus  of  Vanishing  Quan- 
tities, the  Differential  and  Integral  Calculus,  &c.,  ac- 
cording to  the  point  of  view  in  which  it  has  been  con- 
ceived. 

In  order  to  remove  every  foreign  consideration,  I  will 
propose  to  name  it  CALCULUS  OF  INDIRECT  FUNCTIONS,  giv- 
ing to  ordinary  analysis  the  title  of  CALCULUS  OF  DIRECT 
FUNCTIONS.  These  expressions,  which  I  form  essentially 
by  generalizing  and  epitomizing  the  ideas  of  Lagrange, 
are  simply  intended  to  indicate  with  precision  the  true 
general  character  belonging  to  each  of  these  two  forms 
of  analysis. 


68      ANALYSIS,  OR  THE  CALCULUS. 

Having  now  established  the  fundamental  division  of 
mathematical  analysis,  I  have  next  to  consider  separate- 
ly each  of  its  two  parts,  commencing  with  the  Calculus 
of  Direct  Functions,  and  reserving  more  extended  de- 
velopments for  the  different  branches  of  the  Calculus  of 
Indirect  Functions. 


CHAPTER    II. 

ORDINARY  ANALYSIS,    OR   ALGEBRA. 

THE  Calculus  of  direct  Functions,  or  Algebra,  is  (as 
was  shown  at  the  end  of  the  preceding  chapter)  entirely 
sufficient  for  the  solution  of  mathematical  questions,  when 
they  are  so  simple  that  we  can  form  directly  the  equa- 
tions between  the  magnitudes  themselves  which  we  are 
considering,  without  its  being  nece*ssary  to  introduce  in 
their  place,  or  conjointly  with  them,  any  system  of  aux- 
iliary quantities  derived  from  the  first.  It  is  true  that 
in  the  greatest  number  of  important  cases  its  use  re- 
quires to  be  preceded  and  prepared  by  that  of  the  Cal- 
culus of  indirect  Functions,  which  is  intended  to  facili- 
tate the  establishment  of  equations.  But,  although  alge- 
bra has  then  only  a  secondary  office  to  perform,  it  has 
none  the  less  a  necessary  part  in  the  complete  solution 
of  the  question,  so  that  the  Calculus  of  direct  Functions 
must  continue  to  be,  by  its  nature,  the  fundamental  base 
of  all  mathematical  analysis.  We  must  therefore,  before 
going  any  further,  consider  in  a  general  manner  the  logi- 
cal composition  of  this  calculus,  and  the  degree  of  devel- 
opment to  which  it  has  at  the  present  day  arrived. 

Its  Object.  The  final  object  of  this  calculus  being  the 
resolution  (properly  so  called)  of  equations,  that  is,  the 
discovery  of  the  manner  in  which  the  unknown  quan- 
tities are  formed  from  the  known  quantities,  in  accord- 
ance with  the  equations  which  exist  between  them,  it 
naturally  presents  as  many  different  departments  as  we 


70  ORDINARY  ANALYSIS. 

can  conceive  truly  distinct  classes  of  equations.  Its  ap- 
propriate extent  is  consequently  rigorously  indefinite,  the 
number  of  analytical  functions  susceptible  of  entering 
into  equations  being  in  itself  quite  unlimited,  although 
they  are  composed  of  only  a  very  small  number  of  primi- 
tive elements. 

Classification  of  Equations.  The  rational  classifica- 
tion of  equations  must  evidently  be  determined  by  the 
nature  of  the  analytical  elements  of  which  their  numbers 
are  composed ;  every  other  classification  would  be  essen- 
tially arbitrary.  Accordingly,  analysts  begin  by  divid- 
ing equations  with  one  or  more  variables  into  two  princi- 
pal classes,  according  as  they  contain  functions  of  only 
the  first  three  couples  (see  the  table  in  chapter  i.,  page 
51),  or  as  they  include  also  exponential  or  circular  func- 
tions. The  names  of  Algebraic  functions  and  Transcen- 
dental functions,  commonly  given  to  these  two  principal 
groups  of  analytical  elements,  are  undoubtedly  very  in- 
appropriate. But  the  universally  established  division  be- 
tween the  corresponding  equations  is  none  the  less  very 
real  in  this  sense,  that  the  resolution  of  equations  con- 
taining the  functions  called  transcendental  necessarily 
presents  more  difficulties  than  those  of  the  equations 
called  algebraic.  Hence  the  study  of  the  former  is  as 
yet  exceedingly  imperfect,  so  that  frequently  the  resolu- 
tion of  the  most  simple  of  them  is  still  unknown  to  us,^ 
and  our  analytical  methods  have  almost  exclusive  refer- 
ence to  the  elaboration  of  the  latter. 

*  Simple  as  may  seem,  for  example,  the  equation 

ar+6x=cx, 

we  do  not  yet  know  how  to  resolve  it,  which  may  give  some  idea  of  the 
extreme  imperfection  of  this  part  of  algebra. 


ALGEBRAIC  EQUATIONS.  7} 

ALGEBRAIC   EQUATIONS. 

Considering  now  only  these  Algebraic  equations,  we 
must  observe,  in  the  first  place,  that  although  they  may 
often  contain  irrational  functions  of  the  unknown  quan- 
tities as  well  as  rational  functions,  we  can  always,  by 
more  or  less  easy  transformations,  make  the  first  case 
come  under  the  second,  so  that  it  is  with  this  last  that 
analysts  have  had  to  occupy  themselves  exclusively  in 
order  to  resolve  all  sorts  of  algebraic  equations. 

Their 'Classification.  In  the  infancy  of  algebra,  these 
equations  were  classed  according  to  the  number  of  their 
terms.  But  this  classification  was  evidently  faulty,  since 
it  separated  cases  which  were  really  similar,  and  brought 
together  others  which  had  nothing  in  common  besides  this 
unimportant  characteristic.^  It  has  been  retained  only 
for  equations  with  two  terms,  which  are,  in  fact,  capable 
of  being  resolved  in  a  manner  peculiar  to  themselves. 

The  classification  of  equations  by  what  is  called  their 
degrees,  is,  on  the  other  hand,  eminently  natural,  for  this 
distinction  rigorously  determines  the  greater  or  less  dif- 
ficulty of  their  resolution.  This  gradation  is  apparent 
in  the  cases  of  all  the  equations  which  can  be  resolved ; 
but  it  may  be  indicated  in  a  general  manner  independ- 
ently of  the  fact  of  the  resolution.  We  need  only  con- 
sider that  the  most  general  equation  of  each  degree  ne- 
cessarily comprehends  all  those  of  the  different  inferior  de- 
grees, as  must  also  the  formula  which  determines  the  un- 
known quantity.  Consequently,  however  slight  we  may 
suppose  the  difficulty  peculiar  to  the  degree  which  we 

*  The  same  error  was  afterward  committed,  in  the  infancy  of  the  infini- 
tesimal calculus,  in  relation  to  the  integration  of  differential  equations. 


72  ORDINARY  ANALYSIS. 

are  considering,  since  it  is  inevitably  complicated  in  the 
execution  with  those  presented  by  all  the  preceding  de- 
grees, the  resolution  really  offers  more  and  more  obstacles, 
in  proportion  as  the  degree  of  the  equation  is  elevated. 

ALGEBRAIC   RESOLUTION   OF  EQUATIONS. 

Its  Limits.  The  resolution  of  algebraic  equations  is 
as  yet  known  to  us  only  in  the  four  first  degrees,  such 
is  the  increase  of  difficulty  noticed  above.  In  this  re- 
spect, algebra  has  made  no  considerable  progress  since 
the  labours  of  Descartes  and  the  Italian  analyses  of  the 
sixteenth  century,  although  in  the  last  two  centuries 
there  has  been  perhaps  scarcely  a  single  geometer  who 
has  not  busied  himself  in  trying  to  advance  the  resolu- 
tion of  equations.  The  general  equation  of  the  fifth  de- 
gree itself  has  thus  far  resisted  all  attacks. 

The  constantly  increasing  complication  which  the 
formulas  for  resolving  equations  must  necessarily  pre- 
sent, in  proportion  as  the  degree  increases  (the  difficulty 
of  using  the  formula  of  the  fourth  degree  rendering  it  al- 
most inapplicable),  has  determined  analysts  to  renounce, 
by  a  tacit  agreement,  the  pursuit  of  such  researches,  al- 
though they  are  far  from  regarding  it  as  impossible  to 
obtain  the  resolution  of  equations  of  the  fifth  degree,  and 
of  several  other  higher  ones. 

General  Solution.  The  only  question  of  this  kind 
which  would  be  really  of  great  importance,  at  least  in 
its  logical  relations,  would  be  the  general  resolution  of 
algebraic  equations  of  any  degree  whatsoever.  Now, 
the  more  we  meditate  on  this  subject,  the  more  we  are 
led  to  think,  with  Lagrange,  that  it  really  surpasses  the 
scope  of  our  intelligence.  We  must  besides  observe  that 


ALGEBRAIC   RESOLUTION   OF  .EQUATIONS.  73 

the  formula  which  would  express  the  root  of  an  equation 
of  the  mlh  degree  would  necessarily  include  radicals  of 
the  mth  order  (or  functions  of  an  equivalent  multiplici- 
ty), because  of  the  m  determinations  which  it  must  ad- 
mit. Since  we  have  seen,  besides,  that  this  formula 
must  also  embrace,  as  a  particular  case,  that  formula 
which  corresponds  to  every  lower  degree,  it  follows  that 
it  would  inevitably  also  contain  radicals  of  the  next 
lower  degree,  the  next  lower  to  that,  &c.,  so  that,  even 
if  it  were  possible  to  discover  it,  it  would  almost  always 
present  too  great  a  complication  to  be  capable  of  being 
usefully  employed,  unless  we  could  succeed  in  simplify- 
ing it,  at  the  same  time  retaining  all  its  generality,  by 
the  introduction  of  a  new  class  of  analytical  elements  of 
which  we  yet  have  no  idea.  We  have,  then,  reason  to 
believe  that,  without  having  already  here  arrived  at  the 
limits  imposed  by  the  feeble  extent  of  our  intelligence, 
we  should  not  be  long  in  reaching  them  if  we  actively 
and  earnestly  prolonged  tliis  series  of  investigations. 

It  is,  besides,  important  to  observe  that,  even  suppos- 
ing we  had  obtained  the  resolution  of  algebraic  equa- 
tions of  any  degree  whatever,  we  would  still  have  treated 
only  a  very  small  part  of  algebra,  properly  so  called, 
that  is,  of  the  calculus  of  direct  functions,  including  the 
resolution  of  all  the  equations  which  can  be  formed  by 
the  known  analytical  functions. 

Finally,  we  must  remember  that,  by  an  undeniable 
law  of  human  nature,  our  means  for  conceiving  new 
questions  being  much  more  powerful  than  our  resources 
for  resolving  them,  or,  in  other  words,  the  human  mind 
being  much  more  ready  to  inquire  than  to  reason,  we 
shall  necessarily  always  remain  below  the  difficulty,  no 


74  ORDINARY  ANALYSIS. 

matter  to  what  degree  of  development  our  intellectual 
labour  may  arrive.  Thus,  even  though  we  should  some 
day  discover  the  complete  resolution  of  all  the  analytical 
equations  at  present  known,  chimerical  as  the  supposi- 
tion is,  there  can  be  no  doubt  that,  before  attaining  this 
end,  and  probably  even  as  a  subsidiary  means,  we  would 
have  already  overcome  the  difficulty  (a  much  smaller  one, 
though  still  very  great)  of  conceiving  new  analytical  ele- 
ments, the  introduction  of  which  would  give  rise  to  class- 
es of  equations  of  which,  at  present,  we  are  completely 
ignorant ;  so  that  a  similar  imperfection  in  algebraic  sci- 
ence would  be  continually  reproduced,  in  spite  of  the  real 
and  very  important  increase  of  the  absolute  mass  of  our 
knowledge. 

What  we  know  in  Algebra.  In  the  present  condi- 
tion of  algebra,  the  complete  resolution  of  the  equations 
of  the  first  four  degrees,  of  any  binomial  equations,  of 
certain  particular  equations  of  the  higher  degrees,  and  of 
a  very  small  number  of  exponential,  logarithmic,  or  cir- 
cular equations,  constitute  the  fundamental  methods 
which  are  presented  by  the  calculus  of  direct  functions 
for  the  solution  of  mathematical  problems.  But,  limited 
as  these  elements  are,  geometers  have  nevertheless  suc- 
ceeded in  treating,  in  a  truly  admirable  manner,  a  very 
great  number  of  important  questions,  as  we  shall  find  in 
the  course  of  the  volume.  The  general  improvements 
introduced  within  a  century  into  the  total  system  of 
mathematical  analysis,  have  had  for  their  principal  ob- 
ject to  make  immeasurably  useful  this  little  knowledge 
which  we  have,  instead  of  tending  to  increase  it.  This 
result  has  been  so  fully  obtained,  that  most  frequently 
this  calculus  has  no  real  share  in  the  complete  solution 


NUMERICAL  RESOLUTION  OF  EQUATIONS.  75 

of  the  question,  except  by  its  most  simple  parts ;  those 
which  have  reference  to  equations  of  the  two  first  de- 
grees, with  one  or  more  variables. 

NUMERICAL    RESOLUTION    OF    EQUATIONS. 

The  extreme  imperfection  of  algebra,  with  respect  to 
the  resolution  of  equations,  has  led  analysts  to  occupy 
themselves  with  a  new  class  of  questions,  whose  true 
character  should  be  here  noted.  They  have  busied  them- 
selves in  filling  up  the  immense  gap  in  the  resolution  of 
algebraic  equations  of  the  higher  degrees,  by  what  they 
have  named  the  numerical  resolution  of  equations.  Not 
being  able  to  obtain,  in  general,  the  formula  which  ex- 
presses what  explicit  function  of  the  given  quantities  the 
unknown  one  is,  they  have  sought  (in  the  absence  of  this 
kind  of  resolution,  the  only  one  really  algebraic]  to  de- 
termine, independently  of  that  formula,  at  least  the  value 
of  each  unknown  quantity,  for  various  designated  sys- 
tems of  particular  values  attributed  to  the  given  quan- 
tities. By  the  .successive  labours  of  analysts,  this  in- 
complete and  illegitimate  operation,  which  presents  an 
intimate  mixture  of  truly  algebraic  questions  with  others 
which  are  purely  arithmetical,  has  been  rendered  possi- 
ble in  all  cases  for  equations  of  any  degree  and  even  of 
any  form.  The  methods  for  this  which  we  now  possess 
are  sufficiently  general,  although  the  calculations  to  which 
they  lead  are  often  so  complicated  as  to  render  it  almost 
impossible  to  execute  them.  We  have  nothing  else  to 
do,  then,  in  this  part  of  algebra,  but  to  simplify  the  meth- 
ods sufficiently  to  render  them  regularly  applicable,  which 
we  may  hope  hereafter  to  effect.  In  this  condition  of 
the  calculus  of  direct  functions,  we  endeavour,  in  its  ap- 


76  ORDINARY   ANALYSIS. 

plication,  so  to  dispose  the  proposed  questions  as  finally  to 
require  only  this  numerical  resolution  of  the  equations. 
Its  limited  Usefulness.  Valuable  as  is  such  a  re- 
source in  the  absence  of  the  veritable  solution,  it  is  es- 
sential not  to  misconceive  the  true  character  of  these 
methods,  which  analysts  rightly  regard  as  a  very  imper- 
fect algebra.  In  fact,  we  are  far  from  being  always  able 
to  reduce  our  mathematical  questions  to  depend  finally 
upon  only  the  numerical  resolution  of  equations  ;  that 
can  be  done  only  for  questions  quite  isolated  or  truly 
final,  that  is,  for  the  smallest  number.  Most  questions, 
in  fact,  are  only  preparatory,  and  intended  to  serve  as  an 
indispensable  preparation  for  the  solution  of  other  ques- 
tions. Now,  for  such  an  object,  it  is  evident  that  it  is 
not  the  actual  value  of  the  unknown  quantity  which  it 
is  important  to  discover,  but  the  formula,  which  shows 
how  it  is  derived  from  the  other  quantities  under  con- 
sideration. It  is  this  which  happens,  for  example,  in  a 
very  extensive  class  of  cases,  whenever  a  certain  ques- 
tion includes  at  the  same  time  several  unknown  quanti- 
ties. We  have  then,  first  of  all,  to  separate  them.  By 
suitably  employing  the  simple  and  general  method  so 
happily  invented  by  analysts,  and  which  consists  in  re- 
ferring all  the  other  unknown  quantities  to  one  of  them, 
the  difficulty  would  always  disappear  if  we  knew  how  to 
obtain  the  algebraic  resolution  of  the  equations  under 
consideration,  while  the  numerical  solution  would  then 
be  perfectly  useless.  It  is  only  for  want  of  knowing  the 
algebraic  resolution  of  equations  with  a  single  unknown 
quantity,  that  we  are  obliged  to  treat  Elimination  as  a 
distinct  question,  which  forms  one  of  the  greatest  special 
difficulties  of  common  algebra.  Laborious  as  are  the 


NUMERICAL  RESOLUTION  OF  EQUATIONS.  77 

methods  by  the  aid  of  which  we  overcome  this  difficulty, 
they  are  not  even  applicable,  in  an  entirely  general  man- 
ner, to  the  elimination  of  one  unknown  quantity  between 
two  equations  of  any  form  whatever. 

In  the  most  simple  questions,  and  when  we  have  really 
to  resolve  only  a  single  equation  with  a  single  unknown 
quantity,  this  numerical  resolution  is  none  the  less  a 
very  imperfect  method,  even  when  it  is  strictly  sufficient. 
It  presents,  in  fact,  this  serious  inconvenience  of  obliging 
us  to  repeat  the  whole  series  of  operations  for  the  slight- 
est change  which  may  take  place  in  a  single  one  of  the 
quantities  considered,  although  their  relations  to  one  an- 
other remain  unchanged ;  the  calculations  made  for  one 
case  not  enabling  us  to  dispense  with  any  of  those  which 
relate  to  a  case  very  slightly  different.  This  happens  be- 
cause of  our  inability  to  abstract  and  treat  separately 
that  purely  algebraic  part  of  the  question  which  is  com- 
mon to  all  the  cases  which  result  from  the  mere  varia- 
tion of  the  given  numbers. 

According  to  the  preceding  considerations,  the  calcu- 
lus of  direct  functions,  viewed  in  its  present  state,  di- 
vides into  two  very  distinct  branches,  according  as  its 
subject  is  the  algebraic  resolution  of  equations  or  their 
numerical  resolution.  The  first  department,  the  only 
one  truly  satisfactory,  is  unhappily  very  limited,  and  will 
probably  always  remain  so ;  the  second,  too  often  insuf- 
ficient, has,  at  least,  the  advantage  of  a  much  greater 
generality.  The  necessity  of  clearly  distinguishing  these 
two  parts  is  evident,  because  of  the  essentially  different 
object  proposed  in  each,  and  consequently  the  peculiar 
point  of  view  under  which  quantities  are  therein  con- 
sidered. 


7o  ORDINARY   ANALYSIS. 

/  O 

Different  Divisions  of  the  two  Methods  of  Resolu- 
tion.    If,  moreover,  we  consider  these  parts  with  refer- 
ence to  the  different  methods  of  which  each  is  composed, 
we  find  in  their  logical  distribution  an  entirely  different 
arrangement.     In  fact,  the  first  part  must  be  divided 
according  to  the  nature  of  the  equations  which  we  are 
able  to  resolve,  and  independently  of  every  consideration 
relative  to  the  values  of  the  unknown  quantities.      In 
the  second  part,  on  the  contrary,  it  is  not  according  to 
the  degrees  of  the  equations  that  the  methods  are  natu- 
rally distinguished,  since  they  are  applicable  to  equations 
of  any  degree  whatever ;  it  is  according  to  the  numeri- 
cal character  of  the  values  of  the  unknown  quantities ; 
for,  in  calculating  these  numbers  directly,  without  dedu- 
cing them  from  general  formulas,  different  means  would 
evidently  be  employed  when  the  numbers  are  not  suscep- 
tible of  having  their  values  determined  otherwise  than 
by  a  series  of  approximations,  always  incomplete,  or  when 
they  can  be  obtained  with  entire  exactness.      This  dis- 
tinction of  incommensurable  and  of  commensurable  roots, 
which  require  quite  different  principles  for  their  determi- 
nation, important  as  it  is  in  the  numerical  resolution  of 
equations,  is  entirely  insignificant  in  the  algebraic  reso- 
lution, in  which  the  rational  or  irrational  nature  of  the 
numbers  which  are  obtained  is  a  mere  accident  of  the 
calculation,  which  cannot  exercise  any  influence  over  the 
methods  employed ;  it  is,  in  a  word,  a  simple  arithmetical 
consideration.     We  may  say  as  much,  though  in  a  less 
degree,  of  the  division  of  the  commensurable  roots  them- 
selves into  entire  and  fractional.     In  fine,  the  case  is 
the  same,  in  a  still  greater  degree,  with  the  most  gen- 
eral classification  of  roots,  as  real  and  imaginary.      All 


THE    THEORY    OF   EQUATIONS.  79 

these  different  considerations,  which  are  preponderant  as 
to  the  numerical  resolution  of  equations,  and  which  are 
of  no  importance  in  their  algebraic  resolution,  render  more 
and  more  sensible  the  essentially  distinct  nature  of  these 
two  principal  parts  of  algebra. 

THE    THEORY  OF    EQUATIONS. 

These  two  departments,  which  constitute  the  immedi- 
ate object  of  the  calculus  of  direct  functions,  are  subordi- 
nate to  a  third  one,  purely  speculative,  from  which  both 
of  them  borrow  their  most  powerful  resources,  and  which 
has  been  very  exactly  designated  by  the  general  name 
of  Theory  of  Equations,  although  it  as  yet  relates  only 
to  Algebraic  equations.  The  numerical  resolution  of 
equations,  because  of  its  generality,  has  special  need  of 
this  rational  foundation. 

This  last  and  important  branch  of  algebra  is  naturally 
divided  into  two  orders  of  questions,  viz.,  those  which  re- 
fer to  the  composition  of  equations,  and  those  which  con- 
cern their  transformation;  these  latter  having  for  their 
object  to  modify  the  roots  of  an  equation  without  know- 
ing them,  in  accordance  with  any  given  law,  providing 
that  this  law  is  uniform  in  relation  to  all  the  parts.^ 

*  The  fundamental  principle  on  which  reposes  the  theory  of  equations, 
and  which  is  so  frequently  applied  in  all  mathematical  analysis — the  de- 
composition of  algebraic,  rational,  and  entire  functions,  of  any  degree  what- 
ever, into  factors  of  the  first  degree — is  never  employed  except  for  functions 
of  a  single  variable,  without  any  one  having  examined  if  it  ought  to  be  ex- 
tended to  functions  of  several  variables.  The  general  impossibility  of  such 
a  decomposition  is  demonstrated  by  the  author  in  detail,  but  more  properly 
belongs  to  a  special  treatise. 


80  ORDINARY  ANALYSIS. 


THE   METHOD   OF   INDETERMINATE    COEFFICIENTS. 

To  complete  this  rapid  general  enumeration  of  the  dif- 
ferent essential  parts  of  the  calculus  of  direct  functions, 
I  must,  lastly,  mention  expressly  one  of  the  most  fruitful 
and  important  theories  of  algebra  proper,  that  relating 
to  the  transformation  of  functions  into  series  by  the  aid 
of  what  is  called  the  Method  of  indeterminate  Coeffi- 
cients. This  method,  so  eminently  analytical,  and  which 
must  be  regarded  as  one  of  the  most  remarkable  discov- 
eries of  Descartes,  has  undoubtedly  lost  some  of  its  im- 
portance since  the  invention  and  the  development  of  the 
infinitesimal  calculus,  the  place  of  which  it  might  so  hap- 
pily take  in  some  particular  respects.  But  the  increas- 
ing extension  of  the  transcendental  analysis,  although  it 
has  rendered  this  method  much  less  necessary,  has,  on 
the  other  hand,  multiplied  its  applications  and  enlarged 
its  resources  ;  so  that  by  the  useful  combination  between 
the  two  theories,  which  has  finally  been  effected,  the  use 
of  the  method  of  indeterminate  coefficients  has  become 
at  present  much  more  extensive  than  it  was  even  before 
the  formation  of  the  calculus  of  indirect  functions. 

Having  thus  sketched  the  general  outlines  of  algebra 
proper,  I  have  now  to  offer  some  considerations  on  several 
leading  points  in  the  calculus  of  direct  functions,  our 
ideas  of  which  may  be  advantageously  made  more  clear 
by  a  philosophical  examination. 


NEGATIVE  QUANTITIES.  81 


IMAGINARY   QUANTITIES. 

fhe  difficulties  connected  with  several  peculiar  sym- 
W 'i  to  which  algebraic  calculations  sometimes  lead,  and 
especially  to  the  expressions  called  imaginary,  have  been, 
I  think,  much  exaggerated  through  purely  metaphysical 
considerations,  which  have  been  forced  upon  them,  in  the 
place  of  regarding  these  abnormal  results  in  their  true 
point  of  view  as  simple  analytical  facts.  Viewing  them 
thus,  we  readily  see  that,  since  the  spirit  of  mathemat- 
ical analysis  consists  in  considering  magnitudes  in  refer- 
ence to  their  relations  only,  and  without  any  regard  to 
their  determinate  value,  analysts  are  obliged  to  admit  in- 
differently every  kind  of  expression  which  can  be  engen- 
dered by  algebraic  combinations.  The  interdiction  of 
even  one  expression  because  of  its  apparent  singularity 
would  destroy  the  generality  of  their  conceptions.  The 
common  embarrassment  on  this  subject  seems  to  me  to 
proceed  essentially  from  an  unconscious  confusion  be- 
tween the  idea  of  function  and  the  idea  of  value,  or,  what 
comes  to  the  same  thing,  between  the  algebraic  and  the 
arithmetical  point  of  view.  A  thorough  examination 
would  show  mathematical  analysis  to  be  much  more  clear 
in  its  nature  than  even  mathematicians  commonly  sup- 
pose. 

NEGATIVE    QUANTITIES. 

As  to  negative  quantities,  which  have  given  rise  to  so 
many  misplaced  discussions,  as  irrational  as  useless,  we 
must  distinguish  between  their  abstract  signification  and 
their  concrete  interpretation,  which  have  been  almost  al- 
ways confounded  up  to  the  present  day.  Under  the  first 

F 


82  ORDINARY  ANALYSIS. 

point  of  view,  the  theory  of  negative  quantities  can  be 
established  in  a  complete  manner  by  a  single  algebraical 
consideration.  The  necessity  of  admitting  such  expres- 
sions  is  the  same  as  for  imaginary  quantities,  as  above 
indicated ;  and  their  employment  as  an  analytical  arti- 
fice, to  render  the  formulas  more  comprehensive,  is  a 
mechanism  of  calculation  which  cannot  really  give  rise 
to  any  serious  difficulty.  We  may  therefore  regard  the 
abstract  theory  of  negative  quantities  as  leaving  nothing 
essential  to  desire  ;  it  presents  no  obstacles  but  those  in- 
appropriately introduced  by  sophistical  considerations. 

It  is  far  from  being  so,  however,  with  their  concrete 
theory.  This  consists  essentially  in  that  admirable  prop- 
erty of  the  signs  +  and  — ,  of  representing  analytically 
the  oppositions  of  directions  of  which  certain  magnitudes 
are  susceptible.  This  general  theorem  on  the  relation 
of  the  concrete  to  the  abstract  in  mathematics  is  one  of 
the  most  beautiful  discoveries  which  we  owe  to  the  genius 
of  Descartes,  who  obtained  it  as  a  simple  result  of  prop- 
erly directed  philosophical  observation.  A  great  num- 
ber of  geometers  have  since  striven  to  establish  directly 
its  general  demonstration,  but  thus  far  their  efforts  have 
been  illusory.  Their  vain  metaphysical  considerations 
and  heterogeneous  minglings  of  the  abstract  and  the 
concrete  have  so  confused  the  subject,  that  it  becomes 
necessary  to  here  distinctly  enunciate  the  general  fact. 
It  consists  in  this  :  if,  in  any  equation  whatever,  express- 
ing the  relation  of  certain  quantities  which  are  suscepti- 
ble of  opposition  of  directions,  one  or  more  of  those  quan- 
tities come  to  be  reckoned  in  a  direction  contrary  to  that 
which  belonged  to  them  when  the  equation  was  first  es- 
tablished, it  will  not  be  necessary  to  form  directly  a  new 


NEGATIVE   QUANTITIES.  §3 

equation  for  this  second  state  of  the  phenomena  ;  it  will 
suffice  to  change,  in  the  first  equation,  the  sign  of  each 
of  the  quantities  which  shall  have  changed  its  direction  ; 
and  the  equation,  thus  modified,  will  always  rigorously 
coincide  with  that  which  we  would  have  arrived  at  in 
recommencing  to  investigate,  for  this  new  case,  the  an- 
alytical law  of  the  phenomenon.  The  general  theorem 
conavsts  in  this  constant  and  necessary  coincidence.  Now, 
as  yet,  no  one  has  succeeded  in  directly  proving  this  ;  we 
have  assured  ourselves  of  it  only  by  a  great  number  of 
geometrical  and  mechanical  verifications,  which  are,  it 
is  true,  sufficiently  multiplied,  and  especially  sufficiently 
varied,  to  prevent  any  clear  mind  from  having  the  least 
doubt  of  the  exactitude  and  the  generality  of  this  essen- 
tial property,  but  which,  in  a  philosophical  point  of  view, 
do  not  at  all  dispense  with  the  research  for  so  important 
an  explanation.  The  extreme  extent  of  the  theorem  must 
make  us  comprehend  both  the  fundamental  difficulties  of 
this  research  and  the  high  utility  for  the  perfecting  of 
mathematical  science  which  would  belong  to  the  general 
conception  of  this  great  truth.  This  imperfection  of  the- 
ory, however,  has  not  prevented  geometers  from  making 
the  most  extensive  and  the  most  important  use  of  this 
property  in  all  parts  of  concrete  mathematics. 

It  follows  from  the  above  general  enunciation  of  the 
fact,  independently  of  any  demonstration,  that  the  prop- 
erty of  which  we  speak  must  never  be  applied  to  mag- 
nitudes whose  directions  are  continually  varying,  with- 
out giving  rise  to  a  simple  opposition  of  direction ;  in 
that  case,  the  sign  with  which  every  result  of  calculation 
is  necessarily  affected  is  not  susceptible  of  any  concrete 
interpretation,  and  the  attempts  sometimes  made  to  es- 


84  ORDINARY   ANALYSIS. 

tablish  one  are  erroneous.  This  circumstance  occurs, 
among  other  occasions,  in  the  case  of  a  radius  vector  in 
geometry,  and  diverging  forces  in  mechanics. 

PRINCIPLE    OF    HOMOGENEITY. 

A  second  general  theorem  on  the  relation  of  the  con . 
crete  to  the  abstract  is  that  which  is  ordinarily  desig- 
nated under  the  name  of  Principle  of  Homogeneity.  It 
is  undoubtedly  much  less  important  in  its  applications 
than  the  preceding,  but  it  particularly  merits  our  at- 
tention as  having,  by  its  nature,  a  still  greater  extent, 
since  it  is  applicable  to  all  phenomena  without  distinc- 
tion, and  because  of  the  real  utility  which  it  often  pos- 
sesses for  the  verification  of  their  analytical  laws.  I 
can,  moreover,  exhibit  a  direct  and  general  demonstra- 
tion of  it  which  seems  to  me  very  simple.  It  is  founded 
on  this  single  observation,  which  is  self-evident,  that  the 
exactitude  of  every  relation  between  any  concrete  mag- 
nitudes whatsoever  is  independent  of  the  value  of  the 
units  to  which  they  are  referred  for  the  purpose  of  ex- 
pressing them  in  numbers.  For  example,  the  relation 
which  exists  between  the  three  sides  of  a  right-angled 
triangle  is  the  same,  whether  they  are  measured  by  yards, 
or  by  miles,  or  by  inches. 

It  follows  from  this  general  consideration,  that  every 
equation  which  expresses  the  analytical  law  of  any  phe- 
nomenon must  possess  this  property  of  being  in  no  way 
altered,  when  all  the  quantities  which  are  found  in  it 
are  made  to  undergo  simultaneously  the  change  cor- 
responding to  that  which  their  respective  units  would 
experience.  Now  this  change  evidently  consists  in  all 
the  quantities  of  each  sort  becoming  at  once  m  times 


PRINCIPLE   OF    HOMOGENEITY.  Q5 

smaller,  if  the  unit  which  corresponds  to  them  becomes 
m  times  greater,  or  reciprocally.  Thus  every  equation 
which  represents  any  concrete  relation  whatever  must 
possess  this  characteristic  of  remaining  the  same,  when 
we  make  m  times  greater  all  the  quantities  which  it  con- 
tains, and  which  express  the  magnitudes  between  which 
the  relation  exists  ;  excepting  always  the  numbers  which 
designate  simply  the  mutual  ratios  of  these  different 
magnitudes,  and  which  therefore  remain  invariable  du- 
ring the  change  of  the  units.  It  is  this  property  which 
constitutes  the  law  of  Homogeneity  in  its  most  extended 
signification,  that  is,  of  whatever  analytical  functions  the 
equations  may  be  composed. 

But  most  frequently  we  consider  only  the  cases  in 
which  the  functions  are  such  as  are  called  algebraic, 
and  to  which  the  idea  of  degree  is  applicable.  In  this 
case  we  can  give  more  precision  to  the  general  proposi- 
tion by  determining  the  analytical  character  which  must 
be  necessarily  presented  by  the  equation,  in  order  that 
this  property  may  be  verified.  It  is  easy  to  see,  then, 
that,  by  the  modification  just  explained,  all  the  terms  of 
the  first  degree,  whatever  may  be  their  form,  rational  or 
irrational,  entire  or  fractional,  will  become  m  times  great- 
er ;  all  those  of  the  second  degree,  m2  times ;  those  of 
the  third,  m3  times,  &c.  Thus  the  terms  of  the  same  de- 
gree, however  different  may  be  their  composition,  vary- 
ing in  the  same  manner,  and  the  terms  of  different  de- 
grees varying  in  an  unequal  proportion,  whatever  simi- 
larity there  may  be  in  their  composition,  it  will  be  ne- 
cessary, to  prevent  the  equation  from  being  disturbed, 
that  all  the  terms  which  it  contains  should  be  of  the  same 
degree.  It  is  in  this  that  properly  consists  the  ordinary 


gg  ORDINARY  ANALYSIS. 

theorem  of  Homogeneity,  and  it  is  from  this  circum- 
stance that  the  general  law  has  derived  its  name,  which, 
however,  ceases  to  be  exactly  proper  for  all  other  func- 
tions. 

In  order  to  treat  this  subject  in  its  whole  extent,  it  is 
important  to  observe  an  essential  condition,  to  which  at- 
tention must  be  paid  in  applying  this  property  when  the 
phenomenon  expressed  by  the  equation  presents  magni- 
tudes of  different  natures.  Thus  it  may  happen  that 
the  respective  units  are  completely  independent  of  each 
other,  and  then  the  theorem  of  Homogeneity  will  hold 
good,  either  with  reference  to  all  the  corresponding  classes 
of  quantities,  or  with  regard  to  only  a  single  one  or  more 
of  them.  But  it  will  happen  on  other  occasions  that  the 
different  units  will  have  fixed  relations  to  one  another, 
determined  by  the  nature  of  the  question ;  then  it  will 
be  necessary  to  pay  attention  to  this  subordination  of 
the  units  in  verifying  the  homogeneity,  which  will  not 
exist  any  longer  in  a  purely  algebraic  sense,  and  the 
precise  form  of  which  will  vary  according  to  the  nature 
of  the  phenomena.  Thus,  for  example,  to  fix  our  ideas, 
when,  in  the  analytical  expression  of  geometrical  phe- 
nomena, we  are  considering  at  once  lines,  areas,  and  vol- 
umes, it  will  be  necessary  to  observe  that  the  three  cor- 
responding units  are  necessarily  so  connected  with  each 
other  that,  according  to  the  subordination  generally  es- 
tablished in  that  respect,  when  the  first  becomes  m  times 
greater,  the  second  becomes  m*  times,  and  the  third  m3 
times.  It  is  with  such  a  modification  that  homogeneity 
will  exist  in  the  equations,  in  which,  if  they  are  alge- 
braic, we  will  have  to  estimate  the  degree  of  each  term 
by  doubling  the  exponents  of  the  factors  which  corre- 


PRINCIPLE   OF    HOMOGENEITY.  37 

spend  to  areas,  and  tripling  those  of  the  factors  relating 
to  volumes. 


Such  are  the  principal  general  considerations  relating 
to  the  Calculus  of  Direct  Functions.  We  have  now  to 
pass  to  the  philosophical  examination  of  the  Calculus  of 
Indirect  Functions,  the  much  superior  importance  and 
extent  of  which  claim  a  fuller  development. 


CHAPTER    III. 

TRANSCENDENTAL  ANALYSIS-. 
DIFFERENT    MODES    OF    VIEWING    IT. 

WE  determined,  in  the  second  chapter,  the  philosoph- 
ical character  of  the  transcendental  analysis,  in  whatever 
manner  it  may  be  conceived,  considering  only  the  gen- 
eral nature  of  its  actual  destination  as  a  part  of  mathe- 
matical science.  This  analysis  has  been  presented  by 
geometers  under  several  points  of  view,  really  distinct, 
although  necessarily  equivalent,  and  leading  always  to 
identical  results.  They  may  be  reduced  to  three  prin- 
cipal ones ;  those  of  LEIBNITZ,  of  NEWTON,  and  of  LA- 
GRANGE,  of  which  all  the  others  are  only  secondary  mod- 
ifications. In  the  present  state  of  science,  each  of  these 
three  general  conceptions  offers  essential  advantages  which 
pertain  to  it  exclusively,  without  our  having  yet  suc- 
ceeded in  constructing  a  single  method  uniting  all  these 
different  characteristic  qualities.  This  combination  will 
probably  be  hereafter  effected  by  some  method  founded 
upon  the  conception  of  Lagrange  When  that  impor- 
tant philosophical  labour  shall  have  been  accomplished, 
the  study  of  the  other  conceptions  will  have  only  a  his- 
toric interest ;  but,  until  then,  the  science  must  be  con- 
sidered as  in  only  a  provisional  state,  which  requires  the 
simultaneous  consideration  of  all  the  various  modes  of 
viewing  this  calculus.  Illogical  as  may  appear  this  mul- 
tiplicity of  conceptions  of  one  identical  subject,  still, 
without  them  all,  we  could  form  but  a  very  insufficient 


ITS   EARLY   HISTORY.  89 

idea  of  this  analysis,  whether  in  itself,  or  more  especial- 
ly in  relation  to  its  applications.  This  want  of  system 
in  the  most  important  part  of  mathematical  analysis  will 
not  appear  strange  if  we  consider,  on  the  one  hand,  its 
great  extent  and  its  superior  difficulty,  and,  on  the  oth- 
er, its  recent  formation. 

ITS    EARLY    HISTORY. 

If  we  had  to  trace  here  the  systematic  history  of  the 
successive  formation  of  the  transcendental  analysis,  it 
would  be  necessary  previously  to  distinguish  carefully 
from  the  calculus  of  indirect  functions,  properly  so  call- 
ed, the  original  idea  of  the  infinitesimal  method,  which 
can  be  conceived  by  itself,  independently  of  any  calculus. 
We  should  see  that  the  first  germ  of  this  idea  is  found 
in  the  procedure  constantly  employed  by  the  Greek  ge- 
ometers, under  the  name  of  the  Method  of  Exhaustions, 
as  a  means  of  passing  from  the  properties  of  straight  lines 
to  those  of  curves,  and  consisting  essentially  in  substi- 
tuting for  the  curve  the  auxiliary  consideration  of  an  in- 
scribed or  circumscribed  polygon,  by  means  of  which  they 
rose  to  the  curve  itself,  taking  in  a  suitable  manner  the 
limits  of  the  primitive  ratios.  Incontestable  as  is  this 
filiation  of  ideas,  it  would  be  giving  it  a  greatly  exag- 
gerated importance  to  see  in  this  method  of  exhaustions 
the  real  equivalent  of  our  modern  methods,  as  some  ge- 
ometers have  done  ;  for  the  ancients  had  no  logical  and 
general  means  for  the  determination  of  these  limits,  and 
this  was  commonly  the  greatest  difficulty  of  the  ques- 
tion ;  so  that  their  solutions  were  not  subjected  to  ab- 
stract and  invariable  rules,  the  uniform  application  of 
which  would  lead  with  certainty  to  the  knowledge  sought ' 


90  TRANSCENDENTAL   ANALYSIS. 

which  is,  on  the  contrary,  the  principal  characteristic  of 
our  transcendental  analysis.  In  a  word,  there  still  re- 
mained the  task  of  generalizing  the  conceptions  used  by 
the  ancients,  and,  more  especially,  by  considering  it  in  a 
manner  purely  abstract,  of  reducing  it  to  a  complete  sys- 
tem of  calculation,  which  to  them  was  impossible. 

The  first  idea  which  was  produced  in  this  new  direc- 
tion goes  back  to  the  great  geometer  Format,  whom  La- 
grange  has  justly  presented  as  having  blocked  out  the 
direct  formation  of  the  transcendental  analysis  by  his 
method  for  the  determination  of  maxima  and  minima, 
and  for  the  finding  of  tangents,  which  consisted  essen- 
tially in  introducing  the  auxiliary  consideration  of  the 
correlative  increments  of  the  proposed  variables,  incre- 
ments afterward  suppressed  as  equal  to  zero  when  the 
equations  had  undergone  certain  suitable  transforma- 
tions. But,  although  Fermat  was  the  first  to  conceive 
this  analysis  in  a  truly  abstract  manner,  it  was  yet  far 
from  being  regularly  formed  into  a  general  and  distinct 
calculus  having  its  own  notation,  and  especially  freed 
from  the  superfluous  consideration  of  terms  which,  in  the 
analysis  of  Fermat,  were  finally  not  taken  into  the  ac- 
count, after  having  nevertheless  greatly  complicated  all 
the  operations  by  their  presence.  This  is  what  Leibnitz 
so  happily  executed,  half  a  century  later,  after  some  in- 
termediate modifications  of  the  ideas  of  Fermat  intro- 
duced by  Wallis,  and  still  more  by  Barrow ;  and  he  has 
thus  been  the  true  creator  of  the  transcendental  analy- 
sis, such  as  we  now  employ  it.  This  admirable  dis- 
covery was  so  ripe  (like  all  the  great  conceptions  of  the 
human  intellect  at  the  moment  of  their  manifestation), 
that  Newton,  on  his  side,  had  arrived,  at  the  same  time, 


LEIBNITZ  — INFINITESIMALS.  gj 

or  a  little  earlier,  at  a  method  exactly  equivalent,  by 
considering  this  analysis  under  a  very  different  point  of 
view,  which,  although  more  logical  in  itself,  is  really 
less  adapted  to  give  to  the  common  fundamental  method 
all  the  extent  and  the  facility  which  have  been  imparted 
to  it  by  the  ideas  of  Leibnitz.  Finally,  Lagrange,  put- 
ting aside  the  heterogeneous  considerations  which  had 
guided  Leibnitz  and  Newton,  has  succeeded  in  reducing 
the  transcendental  analysis,  in  its  greatest  perfection,  to 
a  purely  algebraic  system,  which  only  wants  more  apti- 
tude for  its  practical  applications. 

After  this  summary  glance  at  the  general  history  of 
the  transcendental  analysis,  we  will  proceed  to  the  dog- 
matic exposition  of  the  three  principal  conceptions,  in  or- 
der to  appreciate  exactly  their  characteristic  properties, 
and  to  show  the  necessary  identity  of  the  methods  which 
are  thence  derived.  Let  us  begin  with  that  of  Leibnitz. 

METHOD  OF  LEIBNITZ. 

Infinitely  small  Elements.  This  consists  in  introdu- 
cing into  the  calculus,  in  order  to  facilitate  the  establish- 
ment of  equations,  the  infinitely  small  elements  of  which 
all  the  quantities,  the  relations  between  which  are  sought, 
are  considered  to  be  composed.  These  elements  or  dif- 
ferentials will  have  certain  relations  to  one  another, 
which  are  constantly  and  necessarily  more  simple  and 
easy  to  discover  than  those  of  the  primitive  quantities,  and 
by  means  of  which  we  will  be  enabled  (by  a  special  calcu- 
lus having  for  its  peculiar  object  the  elimination  of  these 
auxiliary  infinitesimals)  to  go  back  to  the  desired  equa- 
tions, which  it  would  have  been  most  frequently  impos- 
sible to  obtain  directly.  This  indirect  analysis  may  have 


92  TRANSCENDENTAL  ANALYSIS. 

different  degrees  of  indirectness ;  for,  when  there  is  too 
much  difficulty  in  forming  immediately  the  equation  be- 
tween the  differentials  of  the  magnitudes  under  consid- 
eration, a  second  application  of  the  same  general  artifice 
will  have  to  be  made,  and  these  differentials  be  treated, 
in  their  turn,  as  new  primitive  quantities,  and  a  relation 
be  sought  between  their  infinitely  small  elements  (which, 
with  reference  to  the  final  objects  of  the  question,  will  be 
second  differentials),  and  so  on ;  the  same  transforma- 
tion admitting  of  being  repeated  any  number  of  times, 
on  the  condition  of  finally  eliminating  the  constantly  in- 
creasing number  of  infinitesimal  quantities  introduced  as 
auxiliaries. 

A  person  not  yet  familiar  with  these  considerations 
does  not  perceive  at  once  how  the  employment  of  these 
auxiliary  quantities  can  facilitate  the  discovery  of  the 
analytical  laws  of  phenomena ;  for  the  infinitely  small 
increments  of  the  proposed  magnitudes  being  of  the  same 
species  with  them,  it  would  seem  that  their  relations 
should  not  be  obtained  with  more  ease,  inasmuch  as  the 
greater  or  less  value  of  a  quantity  cannot,  in  fact,  exer- 
cise any  influence  on  an  inquiry  which  is  necessarily  in- 
dependent, by  its  nature,  of  every  idea  of  value.  But 
it  is  easy,  nevertheless,  to  explain  very  clearly,  and  in  a 
quite  general  manner,  how  far  the  question  must  be  sim- 
plified by  such  an  artifice.  For  this  purpose,  it  is  ne- 
cessary to  begin  by  distinguishing  different  orders  of  in- 
finitely small  quantities,  a  very  precise  idea  of  which 
may  be  obtained  by  considering  them  as  being  either  the 
successive  powers  of  the  same  primitive  infinitely  small 
quantity,  or  as  being  quantities  which  may  be  regarded 
as  having  finite  ratios  with  these  powers ;  so  that,  to 


THE  INFINITES-IMAL  METHOD.  93 

take  an  example,  the  second,  third,  &c.,  differentials  of 
any  one  variable  are  classed  as  infinitely  small  quanti- 
ties of  the  second  order,  the  third,  &c.,  because  it  is 
easy  to  discover  in  them  finite  multiples  of  the  second, 
third,  &c.,  powers  of  a  certain  first  differential.  These 
preliminary  ideas  being  established,  the  spirit  of  the  in- 
finitesimal analysis  consists  in  constantly  neglecting  the 
infinitely  small  quantities  in  comparison  with  finite  quan- 
tities, and  generally  the  infinitely  small  quantities  of  any 
order  whatever  in  comparison  with  all  those  of  an  in- 
ferior order.  It  is  at  once  apparent  how  much  such  a 
liberty  must  facilitate  the  formation  of  equations  between 
the  differentials  of  quantities,  since,  in  the  place  of  these 
differentials,  we  can  substitute  such  other  elements  as  we 
may  choose,  and  as  will  be  more  simple  to  consider,  only 
taking  care  to  conform  to  this  single  condition,  that  the 
new  elements  differ  from  the  preceding  ones  only  by  quan- 
tities infinitely  small  in  comparison  with  them.  It  is 
thus  that  it  will  be  possible,  in  geometry,  to  treat  curved 
lines  as^  composed  of  an  infinity  of  rectilinear  elements, 
curved  surfaces  as  formed  of  plane  elements,  and,  in  me- 
chanics, variable  motions  as  an  infinite  series  of  uniform 
motions,  succeeding  one  another  at  infinitely  small  inter- 
vals of  time. 

EXAMPLES.  Considering  the  importance  of  this  ad- 
mirable conception,  I  think  that  I  ought  here  to  complete 
the  illustration  of  its  fundamental  character  by  the  sum- 
mary indication  of  some  leading  examples. 

1.  Tangents.  Let  it  be  required  to  determine,  for 
each  point  of  a  plane  curve,  the  equation  of  which  is 
given,  the  direction  o/  its  tangent ;  a  question  whose 
general  solution  was  the  primitive  object  of  the  invent- 


94  TRANSCENDENTAL  ANALYSIS. 

ors  of  the  transcendental  analysis.     We  will  consider  th 
tangent  as  a  secant  joining  two  points  infinitely  near  to 
each  other;  and  then,  designating  by  dy  and  dx  the  in- 
finitely  small  differences  of  the  co-ordinates  of  those  two 
points,  the  elementary  principles  of  geometry  will  imme- 

dy 
diately  give  the  equation  t—-r  for  the  trigonometrical 

tangent  of  the  angle  which  is  made  with  the  axis  of  the 
abscissas  by  the  desired  tangent,  this  being  the  most  sim- 
ple way  of  fixing  its  position  in  a  system  of  rectilinear 
co-ordinates.  This  equation,  common  to  all  curves,  being 
established,  the  question  is  reduced  to  a  simple  analytical 
problem,  which  will  consist  in  eliminating  the  infinitesi- 
mals dx  and  dy,  which  were  introduced  as  auxiliaries,  by 
determining  in  each  particular  case,  by  means  of  the  equa- 
tion of  the  proposed  curve,  the  ratio  of  dy  to  dx,  which  will 
be  constantly  done  by  uniform  and  very  simple  methods. 
2.  Rectification  of  an  Arc.  In  the  second  place,  sup- 
pose that  we  wish  to  know  the  length  of  the  arc  of  any 
curve,  considered  as  a  function  of  the  co-ordinates  of  its  ex- 
tremities. It  would  be  impossible  to  establish  directly  tht 
equation  between  this  arc  s  and  these  co-ordinates,  while 
it  is  easy  to  find  the  corresponding  relation  between  the 
differentials  of  these  different  magnitudes.  The  most  sim- 
ple theorems  of  elementary  geometry  will  in  fact  give  at 
once,  considering  the  infinitely  small  arc  ds  as  a  right 
line,  the  equations 

dsz=dy*+dx2,  or  dsz=dx*+dyz+dzz, 
according  as  the  curve  is  of  single  or  double  curvature. 
In  either  case,  the  question  is  now  entirely  within  the 
domain  of  analysis,  which,  by  the  elimination  of  the  dif- 
ferentials (which  is  the  peculiar  object  of  the  calculus  of 


THE  INFINITESIMAL  METHOD.  95 

indirect  functions),  will  carry  us  back  from  this  relation 
to  that  which  exists  between  the  finite  quantities  them- 
selves under  examination. 

3.  Quadrature  of  a  Curve.     It  would  be  the  same 
with  the  quadrature  of  curvilinear  areas.     If  the  curve  is 
a  plane  one,  and  referred  to  rectilinear  co-ordinates,  we 
will  conceive  the  area  A  comprised  between  this  curve, 
the  axis  of  the  abscissas,  and  two  extreme  co-ordinates, 
to  increase  by  an  infinitely  small  quantity  dA,  as  the  re- 
sult of  a  corresponding  increment  of  the  abscissa.      The 
relation  between  these  two  differentials  can  be  immediate- 
ly obtained  with  the  greatest  facility  by  substituting  for 
the  curvilinear  element  of  the  proposed  area  the  rectangle 
formed  by  the  extreme  ordinate  and  the  element  of  the 
abscissa,  from  which  it  evidently  differs  only  by  an  in- 
finitely small  quantity  of  the  second  order.      This  will  at 
once  give,  whatever  may  be  the  curve,  the  very  simple 
differential  equation 

dA=ydxt 

from  which,  when  the  curve  is  defined,  the  calculus  of 
indirect  functions  will  show  how  to  deduce  the  finite 
equation,  which  is  the  immediate  object  of  the  problem. 

4.  Velocity  in  Variable  Motion.     In  like  manner,  in 
Dynamics,  when  we  desire  to  know  the  expression  for 
the  velocity  acquired  at  each  instant  by  a  body  impress- 
ed with  a  motion  varying  according  to  any  law,  we  will 
consider  the  motion  as  being  uniform  during  an  infinite- 
ly small  element  of  the  time  t,  and  we  will  thus  imme- 
diately form  the  differential  equation  de=vdt,  in  which 
v  designates  the  velocity  acquired  when  the  body  has 
passed  over  the  space  e  ;  and  thence  it  will  be  easy  to 
deduce,  by  simple  and  invariable  analytical  procedures, 


96 


TRANSCENDENTAL   ANALYSIS. 


the  formula  which  would  give  the  velocity  in  each  par- 
ticular motion,  in  accordance  with  the  corresponding  re- 
lation between  the  time  and  the  space ;  or,  reciprocally, 
what  this  relation  would  be  if  the  mode  of  variation  of 
the  velocity  was  supposed  to  be  known,  whether  with  re- 
spect to  the  space  or  to  the  time. 

5.  Distribution  of  Heat.  Lastly,  to  indicate  another 
kind  of  questions,  it  is  by  similar  steps  that  we  are  able, 
in  the  study  of  thermological  phenomena,  according  to 
the  happy  conception  of  M.  Fourier,  to  form  in  a  very 
simple  manner  the  general  differential  equation  which 
expresses  the  variable  distribution  of  heat  in  any  body 
whatever,  subjected  to  any  influences,  by  means  of  the 
single  and  easily-obtained  relation,  which  represents  the 
uniform  distribution  of  heat  in  a  right-angled  parallelo- 
pipedon,  considering  (geometrically)  every  other  body  as 
decomposed  into  infinitely  small  elements  of  a  similar 
form,  and  (thermologically)  the  flow  of  heat  as  constant 
during  an  infinitely  small  element  of  time.  Henceforth, 
all  the  questions  which  can  be  presented  by  abstract  ther- 
mology  will  be  reduced,  as  in  geometry  and  mechanics, 
to  mere  difficulties  of  analysis,  which  will  always  consist 
in  the  elimination  of  the  differentials  introduced  as  aux- 
iliaries to  facilitate  the  establishment  of  the  equations. 

Examples  of  such  different  natures  are  more  than  suf- 
ficient to  give  a  clear  general  idea  of  the  immense  scope 
of  the  fundamental  conception  of  the  transcendental  anal- 
ysis as  formed  by  Leibnitz,  constituting,  as  it  undoubt- 
edly does,  the  most  lofty  thought  to  which  the  human 
mind  has  as  yet  attained. 

It  is  evident  that  this  conception  was  indispensable  to 
complete  the  foundation  of  mathematical  science,  by  en- 


THE   INFINITESIMAL   METHOD  97 

abling  us  to  establish,  in  a  broad  and  fruitful  manner, 
the  relation  of  the  concrete  to  the  abstract.  In  this  re- 
spect it  must  be  regarded  as  the  necessary  complement 
of  the  great  fundamental  idea  of  Descartes  on  the  gen- 
eral analytical  representation  of  natural  phenomena  :  an 
idea  which  did  not  begin  to  be  worthily  appreciated  and 
suitably  employed  till  after  the  formation  of  the  infini- 
tesimal analysis,  without  which  it  could  not  produce, 
even  in  geometry,  very  important  results. 

Generality  of  the  Formulas.  Besides  the  admirable 
facility  which  is  given  by  the  transcendental  analysis  for 
the  investigation  of  the  mathematical  laws  of  all  phe- 
nomena, a  second  fundamental  and  inherent  property,  per- 
haps as  important  as  the  first,  is  the  extreme  generality  of 
the  differential  formulas,  which  express  in  a  single  equa- 
tion each  determinate  phenomenon,  however  varied  the 
subjects  in  relation  to  which  it  is  considered.  Thus  we 
see,  in  the  preceding  examples,  that  a  single  differential 
equation  gives  the  tangents  of  all  curves,  another  their 
rectifications,  a  third  their  quadratures ;  and  in  the  same 
way,  one  invariable  formula  expresses  the  mathematical 
law  of  every  variable  motion  ;  and,  finally,  a  single  equa- 
tion constantly  represents  the  distribution  of  heat  in  any 
body  and  for  any  case.  This  generality,  which  is  so  ex- 
ceedingly remarkable,  and  which  is  for  geometers  the 
basis  of  the  most  elevated  considerations,  is  a  fortunate 
and  necessary  consequence  of  the  very  spirit  of  the  trans- 
cendental analysis,  especially  in  the  conception  of  Leib- 
nitz. Thus  the  infinitesimal  analysis  has  not  only  fur- 
nished a  general  method  for  indirectly  forming  equations 
which  it  would  have  been  impossible  to  discover  in  a  di- 
rect manner,  but  it  has  also  permitted  us  to  consider,  for 

G 


98  TRANSCENDENTAL  ANALYSIS. 

the  mathematical  study  of  natural  phenomena,  a  new 
order  of  more  general  laws,  which  nevertheless  present  a 
clear  and  precise  signification  to  every  mind  habituated 
to  their  interpretation.  By  virtue  of  this  second  charac- 
teristic property,  the  entire  system  of  an  immense  sci- 
ence, such  as  geometry  or  mechanics,  has  been  condensed 
into  a  small  number  of  analytical  formulas,  from  which 
the  human  mind  can  deduce,  by  certain  and  invariable 
rules,  the  solution  of  all  particular  problems. 

Demonstration  of  the  Method.  To  complete  the  gen- 
eral exposition  of  the  conception  of  Leibnitz,  there  re- 
mains to  be  considered  the  demonstration  of  the  logical 
procedure  to  which  it  leads,  and  this,  unfortunately,  is 
the  most  imperfect  part  of  this  beautiful  method. 

In  the  beginning  of  the  infinitesimal  analysis,  the 
most  celebrated  geometers  rightly  attached  more  impor- 
tance to  extending  the  immortal  discovery  of  Leibnitz 
and  multiplying  its  applications  than  to  rigorously  es- 
tablishing the  logical  bases  of  its  operations.  They  con- 
tented themselves  for  a  long  time  by  answering  the  ob- 
jections of  second-rate  geometers  by  the  unhoped-for  so- 
lution of  the  most  difficult  problems  ;  doubtless  persuaded 
that  in  mathematical  science,  much  more  than  in  any 
other,  we  may  boldly  welcome  new  methods,  even  when 
their  rational  explanation  is  imperfect,  provided  they  are 
fruitful  in  results,  inasmuch  as  its  much  easier  and  more 
numerous  verifications  would  not  permit  any  error  to  re- 
main long  undiscovered.  But  this  state  of  things  could 
not  long  exist,  and  it  was  necessary  to  go  back  to  the 
very  foundations  of  the  analysis  of  Leibnitz  in  order  to 
prove,  in  a  perfectly  general  manner,  the  rigorous  exact- 
itude of  the  procedures  employed  in  this  method,  in  spite 


THE    INFINITESIMAL   METHOD.  99 

of  the  apparent  infractions  of  the  ordinary  rules  of  rea- 
soning which  it  permitted. 

Leibnitz,  urged  to  answer,  had  presented  an  explana- 
tion entirely  erroneous,  saying  that  he  treated  infinitely 
small  quantities  as  incomparables,  and  that  he  neglected 
them  in  comparison  with  finite  quantities,  "  like  grains 
of  sand  in  comparison  with  the  sea :"  a  view  which  would 
have  completely  changed  the  nature  of  his  analysis,  by 
reducing  it  to  a  mere  approximative  calculus,  which,  un- 
der this  point  of  view,  would  be  radically  vicious,  since 
it  would  be  impossible  to  foresee,  in  general^  to  what  de- 
gree the  successive  operations  might  increase  these  first 
errors,  which  could  thus  evidently  attain  any  amount. 
Leibnitz,  then,  did  not  see,  except  in  a  very  confused 
manner,  the  true  logical  foundations  of  the  analysis  which 
he  had  created.  His  earliest  successors  limited  them- 
selves, at  first,  to  verifying  its  exactitude  by  showing  the 
conformity  of  its  results,  in  particular  applications,  to 
those  obtained  by  ordinary  algebra  or  the  geometry  of  the 
ancients  ;  reproducing,  according  to  the  ancient  methods, 
so  far  as  they  were  able,  the  solutions  of  some  problems  af- 
ter they  had  been  once  obtained  by  the  new  method,  which 
alone  was  capable  of  discovering  them  in  the  first  place. 

When  this  great  question  was  considered  in  a  more 
general  manner,  geometers,  instead  of  directly  attacking 
the  difficulty,  preferred  to  elude  it  in  some  way,  as  Eu- 
ler  and  D'Alembert,  for  example,  have  done,  by  demon- 
strating the  necessary  and  constant  conformity  of  the 
conception  of  Leibnitz,  viewed  in  all  its  applications, 
with  other  fundamental  conceptions  of  the  transcendental 
analysis,  that  of  Newton  especially,  the  exactitude  of 
which  was  free  from  any  objection.  Such  a  general  veri- 


100  TRANSCENDENTAL   ANALYSIS. 

fication  is  undoubtedly  strictly  sufficient  to  dissipate  any 
uncertainty  as  to  the  legitimate  employment  of  the  anal, 
ysis  of  Leibnitz.  But  the  infinitesimal  method  is  so  im- 
portant— it  offers  still,  in  almost  all  its  applications,  such 
a  practical  superiority  over  the  other  general  concep- 
tions which  have  been  successively  proposed — that  there 
would  be  a  real  imperfection  in  the  philosophical  charac- 
ter of  the  science  if  it  could  not  justify  itself,  and  needed 
to  be  logically  founded  on  considerations  of  another  order, 
which  would  then  cease  to  be  employed. 

It  was,  then,  of  real  importance  to  establish  directly 
and  in  a  general  manner  the  necessary  rationality  of  the 
infinitesimal  method.  After  various  attempts  more  or 
less  imperfect,  a  distinguished  geometer,  Carnot,  present- 
ed at  last  the  true  direct  logical  explanation  of  the  meth- 
od of  Leibnitz,  by  showing  it  to  be  founded  on  the  prin- 
ciple of  the  necessary  compensation  of  errors,  this  being, 
in  fact,  the  precise  and  luminous  manifestation  of  what 
Leibnitz  had  vaguely  and  confusedly  perceived.  Carnot 
has  thus  rendered  the  science  an  essential  service,  al- 
though, as  we  shall  see  towards  the  end  of  this  chapter, 
all  this  logical  scaffolding  of  the  infinitesimal  method, 
properly  so  called,  is  very  probably  susceptible  of  only  a 
provisional  existence,  inasmuch  as  it  is  radically  vicious 
in  its  nature.  Still,  we  should  not  fail  to  notice  the 
general  system  of  reasoning  proposed  by  Carnot,  in  order 
to  directly  legitimate  the  analysis  of  Leibnitz.  Here  is 
the  substance  of  it : 

In  establishing  the  differential  equation  of  a  phenome- 
non, we  substitute,  for  the  immediate  elements  of  the  dif- 
ferent quantities  considered,  other  simpler  infinitesimals, 
which  differ  from  them  infinitely  little  in  comparison 


THE    INFINITESIMAL    METHOD.  1Q1 

with  them  ;  and  this  substitution  constitutes  the  princi- 
pal artifice  of  the  method  of  Leibnitz,  which  without  it 
would  possess  no  real  facility  for  the  formation  of  equa- 
tions. Carnot  regards  such  an  hypothesis  as  really  pro- 
ducing an  error  in  the  equation  thus  obtained,  and  which 
for  this  reason  he  calls  imperfect ;  only,  it  is  clear  that 
this  error  must  be  infinitely  small.  Now,  on  the  other 
hand,  all  the  analytical  operations,  whether  of  differen- 
tiation or  of  integration,  which  are  performed  upon  these 
differential  equations,  in  order  to  raise  them  to  finite 
equations  by  eliminating  all  the  infinitesimals  which 
have  been  introduced  as  auxiliaries,  produce  as  constant- 
ly, by  their  nature,  as  is  easily  seen,  other'  analogous  er- 
rors, so  that  an  exact  compensation  takes  place,  and  the 
final  equations,  in  the  words  of  Carnot,  become  perfect. 
Carnot  views,  as  a  certain  and  invariable  indication  of 
the  actual  establishment  of  this  necessary  compensation, 
the  complete  elimination  of  the  various  infinitely  small 
quantities,  which  is  always,  in  fact,  the  final  object  of 
all  the  operations  of  the  transcendental  analysis ;  for  if 
we  have  committed  no  other  infractions  of  the  general 
rules  of  reasoning  than  those  thus  exacted  by  the  very 
nature  of  the  infinitesimal  method,  the  infinitely  small 
errors  thus  produced  cannot  have  engendered  other  than 
infinitely  small  errors  in  all  the  equations,  and  the  rela- 
tions are  necessarily  of  a  rigorous  exactitude  as  soon  as 
they  exist  between  finite  quantities  alone,  since  the  only 
errors  then  possible  must  be  finite  ones,  while  none  such 
can  have  entered.  All  this  general  reasoning  is  founded 
on  the  conception  of  infinitesimal  quantities,  regarded  as 
indefinitely  decreasing,  while  those  from  which  they  are 
derived  are  regarded  as  fixed. 


102     TRANSCENDENTAL  ANALYSIS. 

Illustration  by  Tangents.  Thus,  to  illustrate  this  ab- 
stract exposition  by  a  single  example,  let  us  take  up  again 
the  question  of  tangents,  which  is  the  most  easy  to  an- 

dy 
alyze  completely.     We  will  regard  the  equation  ^=-r-> 

obtained  above,  as  being  affected  with  an  infinitely  small 
error,  since  it  would  be  perfectly  rigorous  only  for  the 
secant.  Now  let  us  complete  the  solution  by  seeking, 
according  to  the  equation  of  each  curve,  the  ratio  be- 
tween the  differentials  of  the  co-ordinates.  If  we  suppose 
this  equation  to  be  y=ax2,  we  shall  evidently  have 

dy=  2axdx+adxz. 

In  this  formula  we  shall  have  to  neglect  the  term  dxz 
as  an  infinitely  small  quantity  of  the  second  order.  Then 
the  combination  of  the  two  imperfect  equations. 

dy 
t=-r-,  dy=2axdx, 

being  sufficient  to  eliminate  entirely  the  infinitesimals, 
the  finite  result,  t=2ax,  will  necessarily  be  rigorously  cor- 
rect, from  the  effect  of  the  exact  compensation  of  the  two 
errors  committed  ;  since,  by  its  finite  nature,  it  cannot  be 
affected  by  an  infinitely  small  error,  and  this  is,  never- 
theless, the  only  one  which  it  could  have,  according  to 
the  spirit  of  the  operations  which  have  been  executed. 

It  would  be  easy  to  reproduce  in  a  uniform  manner 
the  same  reasoning  with  reference  to  all  the  other  gen- 
eral applications  of  the  analysis  of  Leibnitz. 

This  ingenious  theory  is  undoubtedly  more  subtile  than 
solid,  when  we  examine  it  more  profoundly  ;  but  it  has 
really  no  other  radical  logical  fault  than  that  of  the  in- 
finitesimal method  itself,  of  which  it  is,  it  seems  to  me, 
the  natural  development  and  the  general  explanation,  so 


THE    METHOD   OF   LIMITS.  JQ3 

that  ifc  must  be  adopted  for  as  long  a  time  as  it  shall  be 
thought  proper  to  employ  this  method  directly. 

I  pass  now  to  the  general  exposition  of  the  two  other 
fundamental  conceptions  of  the  transcendental  analysis, 
limiting  myself  in  each  to  its  principal  idea,  the  philo- 
sophical character  of  the  analysis  having  been  sufficiently 
determined  above  in  the  examination  of  the  conception 
of  Leibnitz,  which  I  have  specially  dwelt  upon  because 
it  admits  of  being  most  easily  grasped  as  a  whole,  and 
most  rapidly  described. 

METHOD   OF    NEWTON. 

Newton  has  successively  presented  his  own  method  of 
conceiving  the  transcendental  analysis  under  several  dif- 
ferent forms.  That  which  is  at  present  the  most  com- 
monly adopted  was  designated  by  Newton,  sometimes  un- 
der the  name  of  the  Method  of  prime  and  ultimate  Ra- 
tios, sometimes  under  that  of  the  Method  of  Limits. 

Method  of  Limits.  The  general  spirit  of  the  trans- 
cendental analysis,  from  this  point  of  view,  consists  in 
introducing  as  auxiliaries,  in  the  place  of  the  primitive 
quantities,  or  concurrently  with  them,  in  order  to  facili- 
tate the  establishment  of  equations,  the  limits  of  the  ra- 
tios of  the  simultaneous  increments  of  these  quantities  ; 
or,  in  other  words,  the  final  ratios  of  these  increments ; 
limits  or  final  ratios  which  can  be  easily  shown  to  have 
a  determinate  and  finite  value.  A  special  calculus,  which 
is  the  equivalent  of  the  infinitesimal  calculus,  is  then 
employed  to  pass  from  the  equations  between  these  lim- 
its to  the  corresponding  equations  between  the  primitive 
quantities  themselves. 


104  TRANSCENDENTAL  ANALYSIS. 

The  power  which  is  given  by  such  an  analysis,  of  ex- 
pressing with  more  ease  the  mathematical  laws  of  phe- 
nomena, depends  in  general  on  this,  that  since  the  cal- 
culus applies,  not  to  the  increments  themselves  of  the  pro- 
posed quantities,  but  to  the  limits  of  the  ratios  of  those 
increments,  we  can  always  substitute  for  each  increment 
any  other  magnitude  more  easy  to  consider,  provided  that 
their  final  ratio  is  the  ratio  of  equality,  or,  in  other  words, 
that  the  limit  of  their  ratio  is  unity.  It  is  clear,  indeed, 
that  the  calculus  of  limits  would  be  in  no  way  affected 
by  this  substitution.  Starting  from  this  principle,  we 
find  nearly  the  equivalent  of  the  facilities  offered  by  the 
analysis  of  Leibnitz,  which  are  then  merely  conceived  un- 
der another  point  of  view.  Thus  curves  will  be  regard- 
ed as  the  limits  of  a  series  of  rectilinear  polygons,  varia- 
ble motions  as  the  limits  of  a  collection  of  uniform  mo- 
tions of  constantly  diminishing  durations,  and  so  on. 

EXAMPLES.  1.  Tangents.  Suppose,  for  example,  that 
we  wish  to  determine  the  direction  of  the  tangent  to  a 
curve  ;  we  will  regard  it  as  the  limit  towards  which  would 
tend  a  secant,  which  should  turn  about  the  given  point 
so  that  its  second  point  of  intersection  should  indefinitely 
approach  the  first.  Representing  the  differences  of  the  co- 
ordinates of  the  two  points  by  Ay  and  AX,  we  would  have 
at  each  instant,  for  the  trigonometrical  tangent  of  the  an- 
gle which  the  secant  makes  with  the  axis  of  abscissas, 


from  which,  taking  the  limits,  we  will  obtain,  relatively 
to  the  tangent  itself,  this  general  formula  of  transcen- 
dental analysis,  Ay 


THE  METHOD   OF  LIMITS.  1Q5 

the  characteristic  L  being  employed  to  designate  the  limit. 
The  calculus  of  indirect  functions  will  show  how  to  de- 
duce from  this  formula  in  each  particular  case,  when  the 
equation  of  the  curve  is  given,  the  relation  between  t  and 
x,  by  eliminating  the  auxiliary  quantities  which  have 
been  introduced.  If  we  suppose,  in  order  to  complete  the 
solution,  that  the  equation  of  the  proposed  curve  is  y=  ax2, 
we  shall  evidently  have 


from  which  we  shall  obtain 


Now  it  is  clear  that  the  limit  towards  which  the  second 
number  tends,  in  proportion  as  Ax  diminishes,  is  2ax. 
We  shall  therefore  find,  by  this  method,  t=2ax,  as  we 
obtained  it  for  the  same  case  by  the  method  of  Leibnitz. 
2.  Rectifications.  In  like  manner,  when  the  rectifica- 
tion of  a  curve  is  desired,  we  must  substitute  for  the  in- 
crement of  the  arc  s  the  chord  of  this  increment,  which 
evidently  has  such  a  connexion  with  it  that  the  limit 
of  their  ratio  is  unity  ;  and  then  we  find  (pursuing  in 
other  respects  the  same  plan  as  with  the  method  of  Leib- 
nitz) this  general  equation  of  rectifications  : 


/       ASV  /_Ay\s       /_ 

(  L  —  )  =1+  (  L-L  )  +  (  L 

\      AX/  \     AX/        \ 


_ 

or  —    =  -  L  —    , 

AX/ 

according  as  the  curve  is  plane  or  of  double  curvature. 
It  will  now  be  necessary,  for  each  particular  curve,  to 
pass  from  this  equation  to  that  between  the  arc  and  the 
abscissa,  which  depends  on  the  transcendental  calculus 
properly  so  called. 


TRANSCENDENTAL   ANALYSIS. 

We  could  take  up,  with  the  same  facility,  by  the 
method  of  limits,  all  the  other  general  questions,  the  solu- 
tion of  which  has  been  already  indicated  according  to  the 
infinitesimal  method. 

Such  is,  in  substance,  the  conception  which  Newton 
formed  for  the  transcendental  analysis,  or,  more  precise- 
ly, that  which  Maclaurin  and  D'Alembert  have  presented 
as  the  most  rational  basis  of  that  analysis,  in  seeking  to 
fix  and  to  arrange  the  ideas  of  Newton  upon  that  subject. 

Fluxions  and  Fluents.  Another  distinct  form  under 
which  Newton  has  presented  this  same  method  should  be 
here  noticed,  and  deserves  particularly  to  fix  our  atten- 
tion, as  much  by  its  ingenious  clearness  in  some  cases 
as  by  its  having  furnished  the  notation  best  suited  to  this 
manner  of  viewing  the  transcendental  analysis,  and,  more- 
over, as  having  been  till  lately  the  special  form  of  the  cal- 
culus of  indirect  functions  commonly  adopted  by  the  En- 
glish geometers.  I  refer  to  the  calculus  of  fluxions  and 
of  fluents,  founded  on  the  general  idea  of  velocities. 

To  facilitate  the  conception  of  the  fundamental  idea, 
let  us  consider  every  curve  as  generated  by  a  point  im- 
pressed with  a  motion  varying  according  to  any  law  what- 
ever. The  different  quantities  which  the  curve  can  pre- 
sent, the  abscissa,  the  ordinate,  the  arc,  the  area,  &c., 
will  be  regarded  as  simultaneously  produced  by  successive 
degrees  during  this  motion.  The  velocity  with  which 
each  shall  have  been  described  will  be  called  the  fluxion 
of  that  quantity,  which  will  be  inversely  named  its  flu- 
ent. Henceforth  the  transcendental  analysis  will  con- 
sist, according  to  this  conception,  in  forming  directly  the 
equations  between  the  fluxions  of  the  proposed  quanti- 
ties, in  order  to  deduce  therefrom,  by  a  special  calculus, 


FLUXIONS  AND  FLUENTS.       107 

the  equations  between  the  fluents  themselves.  What 
has  been  stated  respecting  curves  may,  moreover,  evi- 
dently be  applied  to  any  magnitudes  whatever,  regard- 
ed, by  the  aid  of  suitable  images,  as  produced  by  motion. 
It  is  easy  to  understand  the  general  and  necessary 
identity  of  this  method  with  that  of  limits  complicated 
with  the  foreign  idea  of  motion.  In  fact,  resuming  the 
case  of  the  curve,  if  we  suppose,  as  we  evidently  always 
may,  that  the  motion  of  the  describing  point  is  uniform 
in  a  certain  direction,  that  of  the  abscissa,  for  example, 
then  the  fluxion  of  the  abscissa  will  be  constant,  like  the 
element  of  the  time  ;  for  all  the  other  quantities  gener- 
ated, the  motion  cannot  be  conceived  to  be  uniform,  ex- 
cept for  an  infinitely  small  time.  Now  the  velocity  being 
in  general  according  to  its  mechanical  conception,  the 
ratio  of  each  space  to  the  time  employed  in  traversing  it, 
and  this  time  being  here  proportional  to  the  increment  of 
the  abscissa,  it  follows  that  the  fluxions  of  the  ordinate, 
of  the  arc,  of  the  area,  &c.,  are  really  nothing  else  (re- 
jecting the  intermediate  consideration  of  time)  than  the 
final  ratios  of  the  increments  of  these  different  quantities 
to  the  increment  of  the  abscissa.  This  method  of  flux- 
ions and  fluents  is,  then,  in  reality,  only  a  manner  of 
representing,  by  a  comparison  borrowed  from  mechanics, 
the  method  of  prime  and  ultimate  ratios,  which  alone  can 
be  reduced  to  a  calculus.  It  evidently,  then,  offers  the 
same  general  advantages  in  the  various  principal  appli- 
cations of  the  transcendental  analysis,  without  its  being 
necessary  to  present  special  proofs  of  this. 


103  TRANSCENDENTAL   ANALYSIS. 

METHOD    OF    LAGRANGE. 

Derived  Functions.  The  conception  of  Lagrange, 
in  its  admirable  simplicity,  consists  in  representing  the 
transcendental  analysis  as  a  great  algebraic  artifice,  by 
which,  in  order  to  facilitate  the  establishment  of  equa- 
tions, we  introduce,  in  the  place  of  the  primitive  func- 
tions, or  concurrently  with  them,  their  derived  func- 
tions ;  that  is,  according  to  the  definition  of  Lagrange, 
the  coefficient  of  the  first  term  of  the  increment  of  each 
function,  arranged  according  to  the  ascending  powers  of 
the  increment  of  its  variable.  The  special  calculus  of 
indirect  functions  has  for  its  constant  object,  .here  as 
well  as  in  the  conceptions  of  Leibnitz  and  of  Newton,  to 
eliminate  these  derivatives  which  have  been  thus  em- 
ployed as  auxiliaries,  in  order  to  deduce  from  their  rela- 
tions the  corresponding  equations  between  the  primitive 
magnitudes. 

An  Extension  of  ordinary  Analysis.  The  transcen- 
dental analysis  is,  then,  nothing  but  a  simple  though  very 
considerable  extension  of  ordinary  analysis.  Geometers 
have  long  been  accustomed  to  introduce  in  analytical  in- 
vestigations, in  the  place  of  the  magnitudes  themselves 
which  they  wished  to  study,  their  different  powers,  or 
their  logarithms,  or  their  sines,  &c.,  in  order  to  simpli- 
fy the  equations,  and  even  to  obtain  them  more  easily. 
This  successive  derivation  is  an  artifice  of  the  same 
nature,  only  of  greater  extent,  and  procuring,  in  conse- 
quence, much  more  important  resources  for  this  common 
object. 

But,  although  we  can  readily  conceive,  a  priori,  that 
the  auxiliary  consideration  of  these  derivatives  may  fa- 


DERIVED  FUNCTIONS.  109 

cilitate  the  establishment  of  equations,  it  is  not  easy  to 
explain  why  this  must  necessarily  follow  from  this  mode 
of  derivation  rather  than  from  any  other  transformation. 
Such  is  the  weak  point  of  the  great  idea  of  Lagrange. 
The  precise  advantages  of  this  analysis  cannot  as  yet  be 
grasped  in  an  abstract  manner,  but  only  shown  by  con- 
sidering separately  each  principal  question,  so  that  the 
verification  is  often  exceedingly  laborious. 

EXAMPLE.  Tangents.  This  manner  of  conceiving  the 
transcendental  analysis  may  be  best  illustrated  by  its  ap- 
plication to  the  most  simple  of  the  problems  above  exam- 
ined —  that  of  tangents. 

Instead  of  conceiving  the  tangent  as  the  prolongation 
of  the  infinitely  small  element  of  the  curve,  according  to 
the  notion  of  Leibnitz  —  or  as  the  limit  of  the  secants,  ac- 
cording to  the  ideas  of  Newton  —  Lagrange  considers  it, 
according  to  its  simple  geometrical  character,  analogous 
to  the  definitions  of  the  ancients,  to  be  a  right  line  such 
that  no  other  right  line  can  pass  through  the  point  of 
contact  between  it  and  the  curve.  Then,  to  determine 
its  direction,  we  must  seek  the  general  expression  of  its 
distance  from  the  curve,  measured  in  any  direction  what- 
ever —  in  that  of  the  ordinate,  for  example  —  and  dispose 
of  the  arbitrary  constant  relating  to  the  inclination  of  the 
right  line,  which  will  necessarily  enter  into  that  expres- 
sion, in  such  a  way  as  to  diminish  that  separation  as  much 
as  possible.  Now  this  distance,  being  evidently  equal 
to  the  difference  of  the  two  ordinates  of  the  curve  and  of 
the  right  line,  which  correspond  to  the  same  new  abscissa 
x+h.  will  be  represented  by  the  formula 


in  which  £  designates,  as  above,  the  unknown  trigonomet- 


HO  TRANSCENDENTAL   ANALYSIS. 

rical  tangent  of  the  angle  which  the  required  line  makes 
with  the  axis  of  abscissas,  and  f'(x)  the  derived  function 
of  the  ordinate/(a;).  This  being  understood,  it  is  easy 
to  see  that,  by  disposing  of  t  so  as  to  make  the  first  term 
of  the  preceding  formula  equal  to  zero,  we  will  render  the 
interval  between  the  two  lines  the  least  possible,  so  that 
any  other  line  for  which  t  did  not  have  the  value  thus 
determined  would  necessarily  depart  farther  from  the  pro- 
posed  curve.  We  have,  then,  for  the  direction  of  the  tan- 
gent sought,  the  general  expression  t=f'(x),  a  result  ex- 
actly equivalent  to  those  furnished  by  the  Infinitesimal 
Method  and  the  Method  of  Limits.  We  have  yet  to  find 
f'(x)  in  each  particular  curve,  which  is  a  mere  question 
of  analysis,  quite  identical  with  those  which  are  present- 
ed, at  this  stage  of  the  operations,  by  the  other  methods. 

After  these  considerations  upon  the  principal  general 
conceptions,  we  need  not  stop  to  examine  some  other  the- 
ories proposed,  such  as  Euler's  Calculus  of  Vanishing- 
Quantities,  which  are  really  modifications — more  or  less 
important,  and,  moreover,  no  longer  used — of  the  preced- 
ing methods. 

I  have  now  to  establish  the  comparison  and  the  appre- 
ciation of  these  three  fundamental  methods.  Their  per- 
fect and  necessary  conformity  is  first  to  be  proven  in  a 
general  manner. 

FUNDAMENTAL   IDENTITY  OF    THE    THREE    METHODS. 

It  is,  in  the  first  place,  evident  from  what  precedes, 
considering  these  three  methods  as  to  their  actual  des- 
tination, independently  of  their  preliminary  ideas,  that 
they  all  consist  in  the  same  general  logical  artifice,  which 
has  been  characterized  in  the  first  chapter  ;  to  wit,  the 


IDENTITY   OF   ALL   THE   METHODS.        m 

introduction  of  a  certain  system  of  auxiliary  magnitudes, 
having  uniform  relations  to  those  which  are  the  special 
objects  of  the  inquiry,  and  substituted  for  them  expressly 
to  facilitate  the  analytical  expression  of  the  mathemati- 
cal laws  of  the  phenomena,  although  they  have  finally  to 
be  eliminated  by  the  aid  of  a  special  calculus.  It  is 
this  which  has  determined  me  to  regularly  define  the 
transcendental  analysis  as  the  calculus  of  indirect  func- 
tions, in  order  to  mark  its  true  philosophical  character, 
at  the  same  time  avoiding  any  discussion  upon  the  best 
manner  of  conceiving  and  applying  it.  The  general  ef- 
fect of  this  analysis,  whatever  the  method  employed,  is, 
then,  to  bring  every  mathematical  question  much  more 
promptly  within  the  power  of  the  calculus,  and  thus  to 
diminish  considerably  the  serious  difficulty  which  is  usu- 
ally presented  by  the  passage  from  the  concrete  to  the  ab- 
stract. Whatever  progress  we  may  make,  we  can  never 
hope  that  the  calculus  will  ever  be  able  to  grasp  every 
question  of  natural  philosophy,  geometrical,  or  mechani- 
cal, or  thermological,  &c.,  immediately  upon  its  birth, 
which  would  evidently  involve  a  contradiction.  Every 
problem  will  constantly  require  a  certain  preliminary  la- 
bour to  be  performed,  in  which  the  calculus  can  be  of  no 
assistance,  and  which,  by  its  nature,  cannot  be  subject- 
ed to  abstract  and  invariable  rules  ;  it  is  that  which  has 
for  its  special  object  the  establishment  of  equations,  which 
form  the  indispensable  starting  point  of  all  analytical  re- 
searches. But  this  preliminary  labour  has  been  remarka- 
bly simplified  by  the  creation  of  the  transcendental  analy- 
sis, which  has  thus  hastened  the  moment  af  which  the 
solution  admits  of  the  uniform  and  precise  application  of 
general  and  abstract  methods  ;  by  reducing,  in  each  case, 


H2  TRANSCENDENTAL  ANALYSIS. 

this  special  labour  to  the  investigation  of  equations  be- 
tween the  auxiliary  magnitudes ;  from  which  the  calculus 
then  leads  to  equations  directly  referring  to  the  proposed 
magnitudes,  which,  before  this  admirable  conception,  it 
had  been  necessary  to  establish  directly  and  separately. 
Whether  these  indirect  equations  are  differential  equa- 
tions, according  to  the  idea  of  Leibnitz,  or  equations  of 
limits,  conformably  to  the  conception  of  Newton,  or,  lastly, 
derived  equations,  according  to  the  theory  of  Lagrange, 
the  general  procedure  is  evidently  always  the  same. 

But  the  coincidence  of  these  three  principal  methods 
is  not  limited  to  the  common  effect  which  they  produce  ; 
it  exists,  besides,  in  the  very  manner  of  obtaining  it.  In 
fact,  not  only  do  all  three  consider,  in  the  place  of  the 
primitive  magnitudes,  certain  auxiliary  ones,  but,  still 
farther,  the  quantities  thus  introduced  as  subsidiary  are 
exactly  identical  in  the  three  methods,  which  conse- 
quently differ  only  in  the  manner  of  viewing  them.  This 
can  be  easily  shown  by  taking  for  the  general  term  of 
comparison  any  one  of  the  three  conceptions,  especially 
that  of  Lagrange,  which  is  the  most  suitable  to  serve  as 
a  type,  as  being  the  freest  from  foreign  considerations. 
Is  it  not  evident,  by  the  very  definition  of  derived  func- 
tions, that  they  are  nothing  else  than  what  Leibnitz  calls 
differential  coefficients,  or  the  ratios  of  the  differential 
of  each  function  to  that  of  the  corresponding  variable, 
since,  in  determining  the  first  differential,  we  will  be 
obliged,  by  the  very  nature  of  the  infinitesimal  method, 
to  limit  ourselves  to  taking  the  only  term  of  the  incre- 
ment of  tlfe  function  which  contains  the  first  power  of 
the  infinitely  small  increment  of  the  variable  ?  In  the 
same  way,  is  not  the  derived  function,  by  its  nature, 


COMPARATIVE  VALUE  OF  EACH  METHOD.  1  1  3 

likewise  the  necessary  limit  towards  which  tends  the  ra- 
tio between  the  increment  of  the  primitive  function  and 
that  of  its  variable,  in  proportion  as  this  last  indefinitely 
diminishes,  since  it  evidently  expresses  what  that  ratio 
becomes  when  we  suppose  the  increment  of  the  variable 

dii 

to  equal  zero  ?     That  which  is  designated  by  —  in  the 

dx 

method  of  Leibnitz  ;  that  which  ought  to  be  noted  as 

Ay 

L  —  in  that  of  Newton  ;   and  that  which  Lagrange  has 

Ax 


indicated  by  /'(«))  is  constantly  one  same  function,  seen 
from  three  different  points  of  view,  the  considerations 
of  Leibnitz  and  Newton  properly  consisting  in  making 
known  two  general  necessary  properties  of  the  derived 
function.  The  transcendental  analysis,  examined  ab- 
stractedly and  in  its  principle,  is  then  always  the  same, 
whatever  may  be  the  conception  which  is  adopted,  and 
the  procedures  of  the  calculus  of  indirect  functions  are 
necessarily  identical  in  these  different  methods,  which  in 
like  manner  must,  for  any  application  whatever,  lead  con- 
stantly to  rigorously  uniform  results. 

COMPARATIVE  VALUE  OF  THE  THREE  METHODS. 

If  now  we  endeavour  to  estimate  the  comparative  value 
of  these  three  equivalent  conceptions,  we  shall  find  in 
each  advantages  and  inconveniences  which  are  peculiar 
to  it,  and  which  still  prevent  geometers  from  confining 
themselves  to  any  one  of  them,  considered  as  final. 

That  of  Leibnitz.  The  conception  of  Leibnitz  pre- 
sents incontestably,  in  all  its  applications,  a  very  marked 
superiority,  by  leading  in  a  much  more  rapid  manner, 
and  with  much  less  mental  effort,  to  the  formation  of 

H 


114 


TRANSCENDENTAL  ANALYSIS. 


equations  between  the  auxiliary  magnitudes.  It  is  to  its 
use  that  we  owe  the  high  perfection  which  has  been  ac- 
quired by  all  the  general  theories  of  geometry  and  me- 
chanics. Whatever  may  be  the  different  speculative 
opinions  of  geometers  with  respect  to  the  infinitesimal 
method,  in  an  abstract  point  of  view,  all  tacitly  agree  in 
employing  it  by  preference,  as  soon  as  they  have  to  treat 
a  new  question,  in  order  not  to  complicate  the  necessary 
difficulty  by  this  purely  artificial  obstacle  proceeding  from 
a  misplaced  obstinacy  in  adopting  a  less  expeditious  course. 
Lagrange  himself,  after  having  reconstructed  the  trans- 
cendental analysis  on  new  foundations,  has  (with  that 
noble  frankness  which  so  well  suited  his  genius)  rendered 
a  striking  and  decisive  homage  to  the  characteristic  prop- 
erties of  the  conception  of  Leibnitz,  by  following  it  ex- 
clusively in  the  entire  system  of  his  Mecanique  Analy- 
tique.  Such  a  fact  renders  any  comments  unnecessary. 
But  when  we  consider  the  conception  of  Leibnitz  in 
itself  and  in  its  logical  relations,  we  cannot  escape  ad- 
mitting, with  Lagrange,  that  it  is  radically  vicious  in 
this,  that,  adopting  its  own  expressions,  the  notion  of  in- 
finitely small  quantities  is  a  false  idea,  of  which  it  is  in 
fact  impossible  to  obtain  a  clear  conception,  however  we 
may  deceive  ourselves  in  that  matter.  Even  if  we  adopt 
the  ingenious  idea  of  the  compensation  of  errors,  as  above 
explained,  this  involves  the  radical  inconvenience  of  being 
obliged  to  distinguish  in  mathematics  two  classes  of  rea- 
sonings, those  which  are  perfectly  rigorous,  and  those  in 
which  we  designedly  commit  errors  which  subsequently 
have  to  be  compensated.  A  conception  which  leads  to 
such  strange  consequences  is  undoubtedly  very  unsatis- 
factory in  a  logical  point  of  view. 


\ 

COMPARATIVE  VALUE  OF  EACH  METHOD.  H5 

To  say,  as  do  some  geometers,  that  it  is  possible  in 
every  case  to  reduce  the  infinitesimal  method  to  that  of 
limits,  the  logical  character  of  which  is  irreproachable, 
would  evidently  be  to  elude  the  difficulty  rather  than  to 
remove  it ;  besides,  such  a  transformation  almost  entire- 
ly strips  the  conception  of  Leibnitz  of  its  essential  ad- 
vantages of  facility  and  rapidity. 

Finally,  even  disregarding  the  preceding  important 
considerations,  the  infinitesimal  method  would  no  less 
evidently  present  by  its  nature  the  very  serious  defect  of 
breaking  the  unity  of  abstract  mathematics,  by  creating 
a  transcendental  analysis  founded  on  principles  so  differ- 
ent from  those  which  form  the  basis  of  the  ordinary  anal- 
ysis. This  division  of  analysis  into  two  worlds  almost 
entirely  independent  of  each  other,  tends  to  hinder  the 
formation  of  truly  general  analytical  conceptions.  To 
fully  appreciate  the  consequences  of  this,  we  should  have 
to  go  back  to  the  state  of  the  science  before  Lagrange 
had  established  a  general  and  complete  harmony  between 
these  two  great  sections. 

That  of  Newton.  Passing  now  to  the  conception  of 
Newton,  it  is  evident  that  by  its  nature  it  is  not  exposed 
to  the  fundamental  logical  objections  which  are  called 
forth  by  the  method  of  Leibnitz.  The  notion  of  limits 
is,  in  fact,  remarkable  for  its  simplicity  and  its  precision. 
In  the  transcendental  analysis  presented  in  this  manner, 
the  equations  are  regarded  as  exact  from  their  very  ori- 
gin, and  the  general  rules  of  reasoning  are  as  constantly 
observed  as  in  ordinary  analysis.  But,  on  the  other 
hand,  it  is  very  far  from  offering  such  powerful  resour- 
ces for  the  solution  of  problems  as  the  infinitesimal  meth- 
od. The  obligation  which  it  imposes,  of  never  consider- 


HQ  TRANSCENDENTAL   ANALYSIS. 

ing  the  increments  of  magnitudes  separately  and  by  them- 
selves, nor  even  in  their  ratios,  but  only  in  the  limits  of 
those  ratios,  retards  considerably  the  operations  of  the 
mind  in  the  formation  of  auxiliary  equations.  We  may 
even  say  that  it  greatly  embarrasses  the  purely  analyt- 
ical transformations.  Thus  the  transcendental  analysis, 
considered  separately  from  its  applications,  is  far  from  pre- 
senting in  this  method  the  extent  and  the  generality  which 
have  been  imprinted  upon  it  by  the  conception  of  Leib- 
nitz. It  is  very  difficult,  for  example,  to  extend  the  theo- 
ry of  Newton  to  functions  of  several  independent  varia- 
bles. But  it  is  especially  with  reference  to  its  applica- 
tions that  the  relative  inferiority  of  this  theory  is  most 
strongly  marked. 

Several  Continental  geometers,  in  adopting  the  method 
of  Newton  as  the  more  logical  basis  of  the  transcendental 
analysis,  have  partially  disguised  this  inferiority  by  a  seri- 
ous inconsistency,  which  consists  in  applying  to  this  meth- 
od the  notation  invented  by  Leibnitz  for  the  infinitesi- 
mal method,  and  which  is  really  appropriate  to  it  alone. 

dy 

In  designating  by  —  that  which  logically  ought,  in  the 
dx 

Ay 
theory  of  limits,  to  be  denoted  by  L — ,  and  in  extending 

AX 

to  all  the  other  analytical  conceptions  this  displacement 
of  signs,  they  intended,  undoubtedly,  to  combine  the  spe- 
cial advantages  of  the  two  methods ;  but,  in  reality,  they 
have  only  succeeded  in  causing  a  vicious  confusion  be- 
tween them,  a  familiarity  with  which  hinders  the  forma- 
tion of  clear  and  exact  ideas  of  either.  It  would  cer- 
tainly be  singular,  considering  this  usage  in  itself,  that, 
by  the  mere  means  of  signs,  it  could  be  possible  to  effect 


COMPARATIVE  VALUE  OF  EACH  METHOD.  H7 

a  veritable  combination  between  two  theories  so  distinct 
as  those  under  consideration. 

Finally,  the  method  of  limits  presents  also,  though  in 
a  less  degree,  the  greater  inconvenience,  which  I  have 
above  noted  in  reference  to  the  infinitesimal  method,  of 
establishing  a  total  separation  between  the  ordinary  and 
the  transcendental  analysis ;  for  the  idea  of  limits,  though 
clear  and  rigorous,  is  none  the  less  in  itself,  as  Lagrange 
has  remarked,  a  foreign  idea,  upon  which  analytical  theo- 
ries ought  not  to  be  dependent. 

That  of  Lagrange.  This  perfect  unity  of  analysis, 
and  this  purely  abstract  character  of  its  fundamental  no- 
tions, are  found  in  the  highest  degree  in  the  conception 
of  Lagrange,  and  are  found  there  alone ;  it  is,  for  this 
reason,  the  most  rational  and  the  most  philosophical  of 
all.  Carefully  removing  every  heterogeneous  considera- 
tion, Lagrange  has  reduced  the  transcendental  analysis 
to  its  true  peculiar  character,  that  of  presenting  a  very 
extensive  class  of  analytical  transformations,  which  facil- 
itate in  a  remarkable  degree  the  expression  of  the  con- 
ditions of  various  problems.  At  the  same  time,  this  anal- 
ysis is  thus  necessarily  presented  as  a  simple  extension 
of  ordinary  analysis ;  it  is  only  a  higher  algebra.  All  the 
different  parts  of  abstract  mathematics,  previously  so  in- 
coherent, have  from  that  moment  admitted  of  being  con- 
ceived as  forming  a  single  system. 

Unhappily,  this  conception,  which  possesses  such  fun- 
damental properties,  independently  of  its  so  simple  and 
so  lucid  notation,  and  which  is  undoubtedly  destined  to 
become  the  final  theory  of  transcendental  analysis,  be- 
cause of  its  high  philosophical  superiority  over  all  the 
other  methods  proposed,  presents  in  its  present  state  too 


HQ  TRANSCENDENTAL   ANALYSIS. 

many  difficulties  in  its  applications,  as  compaied  with  the 
conception  of  Newton,  and  still  more  with  that  of  Leib- 
nitz, to  be  as  yet  exclusively  adopted.  Lagrange  him- 
self has  succeeded  only  with  great  difficulty  in  rediscov- 
ering, by  his  method,  the  principal  results  already  obtain- 
ed by  the  infinitesimal  method  for  the  solution  of  the  gen- 
eral questions  of  geometry  and  mechanics  ;  we  may  judge 
from  that  what  obstacles  would  be  found  in  treating  in 
the  same  manner  questions  which  were  truly  new  and 
important.  It  is  true  that  Lagrange,  on  several  occa- 
sions, has  shown  that  difficulties  call  forth,  from  men  of 
genius,  superior  efforts,  capable  of  leading  to  the  greatest 
results.  It  was  thus  that,  in  trying  to  adapt  his  method 
to  the  examination  of  the  curvature  of  lines,  which  seemed 
so  far  from  admitting  its  application,  he  arrived  at  that 
beautiful  theory  of  contacts  which  has  so  greatly  per- 
fected that  important  part  of  geometry.  But,  in  spite 
of  such  happy  exceptions,  the  conception  of  Lagrange  has 
nevertheless  remained,  as  a  whole,  essentially  unsuited 
to  applications. 

The  final  result  of  the  general  comparison  which  I 
have  too  briefly  sketched,  is,  then,  as  already  suggested, 
that,  in  order  to  really  understand  the  transcendental  anal- 
ysis, we  should  not  only  consider  it  in  its  principles  ac- 
cording to  the  three  fundamental  conceptions  of  Leib- 
nitz, of  Newton,  and  of  Lagrange,  but  should  besides  ac- 
custom ourselves  to  carry  out  almost  indifferently,  ac- 
cording to  these  three  principal  methods,  and  especially 
according  to  the  first  and  the  last,  the  solution  of  all  im- 
portant questions,  whether  of  the  pure  calculus  of  indirect 
functions  or  of  its  applications.  This  is  a  course  which 
I  could  not  too  strongly  recommend  to  all  those  who  de- 


COMPARATIVE  VALUE  OF  EACH  METHOD.  I  1 9 

sire  to  judge  philosophically  of  this  admirable  creation  of 
the  human  mind,  as  well  as  to  those  who  wish  to  learn 
to  make  use  of  this  powerful  instrument  with  success  and 
with  facility.  In  all  the  other  parts  of  mathematical  sci- 
ence, the  consideration  of  different  methods  for  a  single 
class  of  questions  may  be  useful,  even  independently  of 
its  historical  interest,  but  it  is  not  indispensable ;  here, 
on  the  contrary,  it  is  strictly  necessary. 

Having  determined  with  precision,  in  this  chapter,  the 
philosophical  character  of  the  calculus  of  indirect  func- 
tions, according  to  the  principal  fundamental  conceptions 
of  which  it  admits,  we  have  next  to  consider,  in  the  fol- 
lowing chapter,  the  logical  division  and  the  general  com- 
position of  this  calculus. 


CHAPTER   IV. 

THE  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 
ITS   TWO  FUNDAMENTAL   DIVISIONS. 

THE  calculus  of  indirect  functions,  in  accordance  with 
the  considerations  explained  in  the  preceding  chapter,  is 
necessarily  divided  into  two  parts  (or,  more  properly,  is 
decomposed  into  two  different  calculi  entirely  distinct, 
although  intimately  connected  by  their  nature),  accord, 
ing  as  it  is  proposed  to  find  the  relations  between  the 
auxiliary  magnitudes  (the  introduction  of  which  consti- 
tutes the  general  spirit  of  this  calculus)  by  means  of  the 
relations  between  the  corresponding  primitive  magni- 
tudes ;  or,  conversely,  to  try  to  discover  these  direct 
equations  by  means  of  the  indirect  equations  originally 
established.  Such  is,  in  fact,  constantly  the  double  ob- 
ject of  the  transcendental  analysis. 

These  two  systems  have  received  different  names,  ac- 
cording to  the  point  of  view  under  which  this  analysis 
has  been  regarded.  The  infinitesimal  method,  properly 
so  called,  having  been  the  most  generally  employed  for 
the  reasons  which  have  been  given,  almost  all  geome- 
ters employ  habitually  the  denominations  of  Differen- 
tial Calculus  and  of  Integral  Calculus,  established  by 
Leibnitz,  and  which  are,  in  fact,  very  rational  conse- 
quences of  his  conception.  Newton,  in  accordance  with 
his  method,  named  the  first  the  Calculus  of  Fluxions, 
and  the  second  the  Calculus  of  Fluents,  expressions  which 
were  commonly  employed  in  England.  Finally  follow- 


ITS   TWO   FUNDAMENTAL   DIVISIONS.    1^1 

ing  the  eminently  philosophical  theory  founded  by  La- 
grange,  one  would  be  called  the  Calculus  of  Derived 
Functions,  and  the  other  the  Calculus  of  Primitive 
Functions.  I  will  continue  to  make  use  of  the  terms  of 
Leibnitz,  as  being  more  convenient  for  the  formation  of 
secondary  expressions,  although  I  ought,  in  accordance 
with  the  suggestions  made  in  the  preceding  chapter,  to 
employ  concurrently  all  the  different  conceptions,  ap- 
proaching as  nearly  as  possible  to  that  of  Lagrange. 

THEIR  RELATIONS  TO  EACH  OTHER. 

The  differential  calculus  is  evidently  the  logical  ba- 
sis of  the  integral  calculus ;  for  we  do  not  and  cannot 
know  how  to  integrate  directly  any  other  differential  ex- 
pressions than  those  produced  by  the  differentiation  of 
the  ten  simple  functions  which  constitute  the  general  ele- 
ments of  our  analysis.  The  art  of  integration  consists, 
then,  essentially  in  bringing  all  the  other  cases,  as  far  as 
is  possible,  to  finally  depend  on  only  this  small  number 
of  fundamental  integrations. 

In  considering  the  whole  body  of  the  transcendental 
analysis,  as  I  have  characterized  it  in  the  preceding  chap- 
ter, it  is  not  at  first  apparent  what  can  be  the  peculiar 
utility  of  the  differential  calculus,  independently  of  this 
necessary  relation  with  the  integral  calculus,  which  seems 
as  if  it  must  be,  by  itself,  the  only  one  directly  indispen- 
sable. In  fact,  the  elimination  of  the  infinitesimals  or 
of  the  derivatives,  introduced  as  auxiliaries  to  facilitate 
the  establishment  of  equations,  constituting,  as  we  have 
seen,  the  final  and  invariable  object  of  the  calculus  of  in- 
direct functions,  it  is  natural  to  think  that  the  calculus 
which  teaches  how  to  deduce  from  the  equations  between 


122  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

these  auxiliary  magnitudes,  those  which  exist  between  the 
primitive  magnitudes  themselves,  ought  strictly  to  suffice 
for  the  general  wants  of  the  transcendental  analysis  with- 
out  our  perceiving,  at  the  first  glance,  what  special  and 
constant  part  the  solution  of  the  inverse  question  can 
have  in  such  an  analysis.  It  would  be  a  real  error,  though 
a  common  one,  to  assign  to  the  differential  calculus,  in  or- 
der to  explain  its  peculiar,  direct,  and  necessary  influence, 
the  destination  of  forming  the  differential  equations,  from 
which  the  integral  calculus  then  enables  us  to  arrive  at 
the  finite  equations ;  for  the  primitive  formation  of  dif- 
ferential equations  is  not  and  cannot  be,  properly  speak- 
ing, the  object  of  any  calculus,  since,  on  the  contrary,  it 
forms  by  its  nature  the  indispensable  starting  point  of  any 
calculus  whatever.  How,  in  particular,  could  the  differ- 
ential calculus,  which  in  itself  is  reduced  to  teaching  the 
means  of  differentiating  the  different  equations,  be  a 
general  procedure  for  establishing  them  ?  That  which 
in  every  application  of  the  transcendental  analysis  really 
facilitates  the  formation  of  equations,  is  the  infinitesimal 
method,  and  not  the  infinitesimal  calculus,  which  is  per- 
fectly distinct  from  it,  although  it  is  its  indispensable  com- 
plement. Such  a  consideration  would,  then,  give  a  false 
idea  of  the  special  destination  which  characterizes  the  dif- 
ferential calculus  in  the  general  system  of  the  transcen- 
dental analysis. 

But  we  should  nevertheless  very  imperfectly  conceive 
the  real  peculiar  importance  of  this  first  branch  of  the 
calculus  of  indirect  functions,  if  we  saw  in  it  only  a  sim- 
ple preliminary  labour,  having  no  other  general  and  es- 
sential object  than  to  prepare  indispensable  foundations 
for  the  integral  calculus.  As  the  ideas  on  this  matter 


THEIR  MUTUAL  RELATIONS.      1£3 

are  generally  confused,  I  think  that  I  ought  here  to  ex- 
plain in  a  summary  manner  this  important  relation  as  I 
view  it,  and  to  show  that  in  every  application  of  the 
transcendental  analysis  a  primary,  direct,  and  necessary 
part  is  constantly  assigned  to  the  differential  calculus. 

1.  Use  of  the  Differential  Calculus  as  preparatory 
to  that  of  the  Integral.  In  forming  the  differential  equa- 
tions of  any  phenomenon  whatever,  it  is  very  seldom  that 
we  limit  ourselves  to  introduce  differentially  only  those 
magnitudes  whose  relations  are  sought.  To  impose  that 
condition  would  be  to  uselessly  diminish  the  resources 
presented  by  the  transcendental  analysis  for  the  expres- 
sion of  the  mathematical  laws  of  phenomena.  Most  fre- 
quently we  introduce  into  the  primitive  equations,  through 
their  differentials,  other  magnitudes  whose  relations  are 
already  known  or  supposed  to  be  so,  and  without  the 
consideration  of  which  it  would  be  frequently  impossible 
to  establish  equations.  Thus,  for  example,  in  the  gen- 
eral problem  of  the  rectification  of  curves,  the  differen- 
tial equation, 

ds2=dyz+dx2,  or  dsz=dxz+dyz-\-dzz^ 
is  not  only  established  between  the  desired  function  s  and 
the  independent  variable  z,  to  which  it  is  referred,  but,  at 
the  same  time,  there  have  been  introduced,  as  indispen- 
sable intermediaries,  the  differentials  of  one  or  two  other 
functions,  y  and  z,  which  are  among  the  data  of  the 
problem ;  it  would  not  have  been  possible  to  form  directly 
the  equation  between  ds  and  dx,  which  would,  besides, 
be  peculiar  to  each  curve  considered.  It  is  the  same  for 
most  questions.  Now  in  these  cases  it  is  evident  that 
the  differential  equation  is  not  immediately  suitable  for 
integration.  It  is  previously  necessary  that  the  differ- 


124  DIFFERENTIAL  AND  INTEGRAL  CALCULUS 

entials  of  the  functions  supposed  to  be  known,  which 
have  been  employed  as  intermediaries,  should  be  entirely 
eliminated,  in  order  that  equations  may  be  obtained  be- 
tween the  differentials  of  the  functions  which  alone  are 
sought  and  those  of  the  really  independent  variables,  af- 
ter which  the  question  depends  on  only  the  integral  cal- 
culus. Now  this  preparatory  elimination  of  certain  dif- 
ferentials, in  order  to  reduce  the  infinitesimals  to  the 
smallest  number  possible,  belongs  simply  to  the  differ- 
ential calculus  ;  for  it  must  evidently  be  done  by  deter- 
mining, by  means  of  the  equations  between  the  func- 
tions supposed  to  be  known,  taken  as  intermediaries,  the 
relations  of  their  differentials,  which  is  merely  a  question 
of  differentiation.  Thus,  for  example,  in  the  case  of  rec- 
tifications, it  will  be  first  necessary  to  calculate  dy,  or  dy 
and  dz,  by  differentiating  the  equation  or  the  equations 
of  each  curve  proposed  ;  after  eliminating  these  expres- 
sions, the  general  differential  formula  above  enunciated 
will  then  contain  only  ds  and  dx  ;  having  arrived  at  this 
point,  the  elimination  of  the  infinitesimals  can  be  com- 
pleted only  by  the  integral  calculus. 

Such  is,  then,  the  general  office  necessarily  belonging 
to  the  differential  calculus  in  the  complete  solution  of  the 
questions  which  exact  the  employment  of  the  transcen- 
dental analysis ;  to  produce,  as  far  as  is  possible,  the  elim- 
ination of  the  infinitesimals,  that  is,  to  reduce  in  each 
case  the  primitive  differential  equations  so  that  they  shall 
contain  only  the  differentials  of  the  really  independent 
variables,  and  those  of  the  functions  sought,  by  causing 
to  disappear,  by  elimination,  the  differentials  of  all  the 
other  known  functions  which  may  have  been  taken  as  in- 
termediaries at  the  time  of  the  formation  of  the  differ- 


ITS    TWO    FUNDAMENTAL   DIVISIONS.     J25 

ential  equations  of  the  problem  which  is  under  consid- 
eration. 

2.  Employment  of  the  Differential  Calculus  alone. 
For  certain  questions,  which,  although  few  in  number, 
have  none  the  less,  as  we  shall  see  hereafter,  a  very  great 
importance,  the  magnitudes  which  are  sought  enter  di- 
rectly, and  not  by  their  differentials,  into  the  primitive 
differential  equations,  which  then  contain  differentially 
only  the  different  known  functions  employed  as  interme- 
diaries, in  accordance  with  the  preceding  explanation. 
These  cases  are  the  most  favourable  of  all ;  for  it  is  evi- 
dent that  the  differential  calculus  is  then  entirely  suffi- 
cient for  the  complete  elimination  of  the  infinitesimals, 
without  the  question  giving  rise  to  any  integration.    This 
is  what  occurs,  for  example,  in  the  problem  of  tangents 
in  geometry  ;   in  that  of  velocities  in  mechanics,  &c. 

3.  Employment  of  the  Integral  Calculus  alone.     Fi- 
nally, some  other  questions,  the  number  of  which  is  also 
very  small,  but  the  importance  of  which  is  no  less  great, 
present  a  second  exceptional  case,  which  is  in  its  nature 
exactly  the  converse  of  the  preceding.      They  are  those 
in  which  the  differential  equations  are  found  to  be  im- 
mediately ready  for  integration,  because  they  contain,  at 
their  first  formation,  only  the  infinitesimals  which  relate 
to  the  functions  sought,  or  to  the  really  independent  va- 
riables, without  its  being  necessary  to  introduce,  differ- 
entially, other  functions  as  intermediaries.     If  in  these 
new  cases  we  introduce  these  last  functions,  since,  by  hy- 
pothesis, they  will  enter  directly  and  not  by  their  differ- 
entials, ordinary  algebra  will  suffice  to  eliminate  them, 
and  to  bring  the  question  to  depend  on  only  the  integral 
calculus.      The  differential  calculus  will  then  have  no 


126  DIFFERENTIAL  AND   INTEGRAL   CALCULUS. 

special  part  in  the  complete  solution  of  the  problem,  which 
will  depend  entirely  upon  the  integral  calculus.  The 
general  question  of  quadratures  offers  an  important  ex- 
ample of  this,  for  the  differential  equation  being  then 
dA.=ydx,  will  become  immediately  fit  for  integration  as 
soon  as  we  shall  have  eliminated,  by  means  of  the  equa- 
tion of  the  proposed  curve,  the  intermediary  function  y, 
which  does  not  enter  into  it  differentially.  The  same 
circumstances  exist  in  the  problem  of  cubatures,  and  in 
some  others  equally  important. 

Three  classes  of  Questions  hence  resulting-.  As  a 
general  result  of  the  previous  considerations,  it  is  then 
necessary  to  divide  into  three  classes  the  mathematical 
questions  which  require  the  use  of  the  transcendental 
analysis ;  the  first  class  comprises  the  problems  suscep- 
tible of  being  entirely  resolved  by  means  of  the  differen- 
tial calculus  alone,  without  any  need  of  the  integral  cal- 
culus ;  the  second,  those  which  are,  on  the  contrary,  en- 
tirely dependent  upon  the  integral  calculus,  without  the 
differential  calculus  having  any  part  in  their  solution  ; 
lastly,  in  the  third  and  the  most  extensive,  which  con- 
stitutes the  normal  case,  the  two  others  being  only  ex- 
ceptional, the  differential  and  the  integral  calculus  have 
each  in  their  turn  a  distinct  and  necessary  part  in  the 
complete  solution  of  the  problem,  the  former  making  the 
primitive  differential  equations  undergo  a  preparation 
which  is  indispensable  for  the  application  of  the  latter. 
Such  are  exactly  their  general  relations,  of  which  too 
indefinite  and  inexact  ideas  are  generally  formed. 

Let  us  now  take  a  general  survey  of  the  logical  com- 
position of  each  calculus,  beginning  with  the  differential. 


THE   DIFFERENTIAL    CALCULUS.          J27 


THE    DIFFERENTIAL    CALCULUS. 

In  the  exposition  of  the  transcendental  analysis,  it  is 
customary  to  intermingle  with  the  purely  analytical  part 
(which  reduces  itself  to  the  treatment  of  the  abstract 
principles  of  differentiation  and  integration)  the  study  of 
its  different  principal  applications,  especially  those  which 
concern  geometry.  This  confusion  of  ideas,  which  is  a 
consequence  of  the  actual  manner  in  which  the  science 
has  been  developed,  presents,  in  the  dogmatic  point  of 
view,  serious  inconveniences  in  this  respect,  that  it  makes 
it  difficult  properly  to  conceive  either  analysis  or  geom- 
etry. Having  to  consider  here  the  most  rational  co-or- 
dination which  is  possible,  I  shall  include,  in  the  follow- 
ing sketch,  only  the  calculus  of  indirect  functions  prop- 
erly so  called,  reserving  for  the  portion  of  this  volume 
which  relates  to  the  philosophical  study  of  concrete  math- 
ematics the  general  examination  of  its  great  geometri- 
cal and  mechanical  applications. 

Two  Cases  :  explicit  and  implicit  Functions.  The 
fundamental  division  of  the  differential  calculus,  or  of 
the  general  subject  of  differentiation,  consists  in  distin- 
guishing two  cases,  according  as  the  analytical  functions 
which  are  to  be  differentiated  are  explicit  or  implicit ; 
from  which  flow  two  parts  ordinarily  designated  by  the 
names  of  differentiation  of  formulas  and  differentiation 
of  equations.  It  is  easy  to  understand,  a  priori,  the 
importance  of  this  classification.  In  fact,  such  a  dis- 
tinction would  be  illusory  if  the  ordinary  analysis  was 
perfect ;  that  is,  if  we  knew  how  to  resolve  all  equations 
algebraically,  for  then  it  would  be  possible  to  render 
every  implicit  function  explicit ;  and,  by  differentiating 


128   DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

it  in  that  state  alone,  the  second  part  of  the  differential 
calculus  would  be  immediately  comprised  in  the  first, 
without  giving  rise  to  any  new  difficulty.  But  the  al- 
gebraical resolution  of  equations  being,  as  we  have  seen, 
still  almost  in  its  infancy,  and  as  yet  impossible  for  most 
cases,  it  is  plain  that  the  case  is  very  different,  since 
we  have,  properly  speaking,  to  differentiate  a  function 
without  knowing  it,  although  it  is  determinate.  The 
differentiation  of  implicit  functions  constitutes  then,  by 
its  nature,  a  question  truly  distinct  from  that  presented 
by  explicit  functions,  and  necessarily  more  complicated. 
It  is  thus  evident  that  we  must  commence  with  the  dif- 
ferentiation of  formulas,  and  reduce  the  differentiation 
of  equations  to  this  primary  case  by  certain  invariable 
analytical  considerations,  which  need  not  be  here  men- 
tioned. 

These  two  general  cases  of  differentiation  are  also  dis- 
tinct in  another  point  of  view  equally  necessary,  and  too 
important  to  be  left  unnoticed.  The  relation  which  is 
obtained  between  the  differentials  is  constantly  more  in- 
direct, in  comparison  with  that  of  the  finite  quantities, 
in  the  differentiation  of  implicit  functions  than  in  that 
of  explicit  functions.  We  know,  in  fact,  from  the  con- 
siderations presented  by  Lagrange  on  the  general  forma- 
tion of  differential  equations,  that,  on  the  one  hand,  the 
same  primitive  equation  may  give  rise  to  a  greater  or 
less  number  of  derived  equations  of  very  different  forms, 
although  at  bottom  equivalent,  depending  upon  which  of 
the  arbitrary  constants  is  eliminated,  which  is  not  the 
case  in  the  differentiation  of  explicit  formulas;  and 
that,  on  the  other  hand,  the  unlimited  system  of  the 
different  primitive  equations,  which  correspond  to  the 


THE  DIFFERENTIAL  CALCULUS.          129 

same  derived  equation,  presents  a  much  more  profound 
analytical  variety  than  that  of  the  different  functions, 
which  admit  of  one  same  explicit  differential,  and  which 
are  distinguished  from  each  other  only  by  a  constant 
term.  Implicit  functions  must  therefore  be  regarded  as 
being  in  reality  still  more  modified  by  differentiation 
than  explicit  functions.  We  shall  again  meet  with  this 
consideration  relatively  to  the  integral  calculus,  where 
it  acquires  a  preponderant  importance. 

Two  Sub-cases :  A  single  Variable  or  several  Varia- 
bles. Each  of  the  two  fundamental  parts  of  the  Differ- 
ential Calculus  is  subdivided  into  two  very  distinct  theo- 
ries, according  as  we  are  required  to  differentiate  func- 
tions of  a  single  variable  or  functions  of  several  inde- 
pendent variables.  This  second  case  is,  by  its  nature, 
quite  distinct  from  the  first,  and  evidently  presents  more 
complication,  even  in  considering  only  explicit  functions, 
and  still  more  those  which  are  implicit.  As  to  the  rest, 
one  of  these  cases  is  deduced  from  the  other  in  a  gen- 
eral manner,  by  the  aid  of  an  invariable  and  very  simple 
principle,  which  consists  in  regarding  the  total  differen- 
tial of  a  function  which  is  produced  by  the  simultaneous 
increments  of  the  different  independent  variables  which 
it  contains,  as  the  sum  of  the  partial  differentials  which 
would  be  produced  by  the  separate  increment  of  each 
variable  in  turn,  if  all  the  others  were  constant.  It  is 
necessary,  besides,  carefully  to  remark,  in  connection 
with  this  subject,  a  new  idea  which  is  introduced  by 
the  distinction  of  functions  into  those  of  one  variable 
and  of  several ;  it  is  the  consideration  of  these  different 
special  derived  functions,  relating  to  each  variable  sep- 
arately, and  the  number  of  which  increases  more  and 

I 


130  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

more  in  proportion  as  the  order  of  the  derivation  becomes 
higher,  and  also  when  the  variables  become  more  nu- 
merous. It  results  from  this  that  the  differential  rela- 
tions belonging  to  functions  of  several  variables  are,  by 
their  nature,  both  much  more  indirect,  and  especially 
much  more  indeterminate,  than  those  relating  to  func- 
tions of  a  single  variable.  This  is  most  apparent  in  the 
case  of  implicit  functions,  in  which,  in  the  place  of  the 
simple  arbitrary  constants  which  elimination  causes  to 
disappear  when  we  form  the  proper  differential  equations 
for  functions  of  a  single  variable,  it  is  the  arbitrary  func- 
tions of  the  proposed  variables  which  are  then  elimi- 
nated ;  whence  must  result  special  difficulties  when  these 
equations  come  to  be  integrated. 

Finally,  to  complete  this  summary  sketch  of  the  dif- 
ferent essential  parts  of  the  differential  calculus  proper, 
I  should  add,  that  in  the  differentiation  of  implicit  func- 
tions, whether  of  a  single  variable  or  of  several,  it  is  ne- 
cessary to  make  another  distinction  ;  that  of  the  case  in 
which  it  is  required  to  differentiate  at  once  different 
functions  of  this  kind,  combined  in  certain  primitive 
equations,  from  that  in  which  all  these  functions  are 
separate. 

The  functions  are  evidently,  in  fact,  still  more  im- 
plicit in  the  first  case  than  in  the  second,  if  we  consider 
that  the  same  imperfection  of  ordinary  analysis,  which 
forbids  our  converting  every  implicit  function  into  an 
equivalent  explicit  function,  in  like  manner  renders  us 
unable  to  separate  the  functions  which  enter  simulta- 
neously  into  any  system  of  equations.  It  is  then  ne- 
cessary to  differentiate,  not  only  without  knowing  how 
to  resolve  the  primitive  equations,  but  even  without  be- 


THE  DIFFERENTIAL   CALCULUS.         ^3^ 

ing  able  to  effect  the  proper  eliminations  among  them, 
thus  producing  a  new  difficulty. 

Reduction  of  the  whole  to  the  Differentiation  of  the 
ten  elementary  Functions.  Such,  then,  are  the  natural 
connection  and  the  logical  distribution  of  the  different 
principal  theories  which  compose  the  general  system  of 
differentiation.  Since  the  differentiation  of  implicit 
functions  is  deduced  from  that  of  explicit  functions  by 
a  single  constant  principle,  and  the  differentiation  of 
functions  of  several  variables  is  reduced  by  another  fixed 
principle  to  that  of  functions  of  a  single  variable,  the 
whole  of  the  differential  calculus  is  finally  found  to  rest 
upon  the  differentiation  of  explicit  functions  with  a  sin- 
gle variable,  the  only  one  which  is  ever  executed  direct- 
ly. Now  it  is  easy  to  understand  that  this  first  theory, 
the  necessary  basis  of  the  entire  system,  consists  simply 
in  the  differentiation  of  the  ten  simple  functions,  which 
are  the  uniform  elements  of  all  our  analytical  combina- 
tions, and  the  list  of  which  has  been  given  in  the  first 
chapter,  on  page  51 ;  for  the  differentiation  of  compound 
functions  is  evidently  deduced,  in  an  immediate  and  ne- 
cessary manner,  from  that  of  the  simple  functions  which 
compose  them.  It  is,  then,  to  the  knowledge  of  these 
ten  fundamental  differentials,  and  to  that  of  the  two  gen- 
eral principles  just  mentioned,  which  bring  under  it  all 
the  other  possible  cases,  that  the  whole  system  of  differ- 
entiation is  properly  reduced.  We  see,  by  the  combina- 
tion of  these  different  considerations,  how  simple  and 
how  perfect  is  the  entire  system  of  the  differential  cal- 
culus. It  certainly  constitutes,  in  its  logical  relations, 
the  most  interesting  spectacle  which  mathematical  analy- 
sis can  present  to  our  understanding. 


132  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Transformation  of  derived  Functions  for  new  Varia. 
bles.  The  general  sketch  which  I  have  just  summarily 
drawn  would  nevertheless  present  an  important  deficien- 
cy, if  I  did  not  here  distinctly  indicate  a  final  theory, 
which  forms,  by  its  nature,  the  indispensable  complement 
of  the  system  of  differentiation.  It  is  that  which  has 
for  its  object  the  constant  transformation  of  derived  func- 
tions, as  a  result  of  determinate  changes  in  the  inde- 
pendent variables,  whence  results  the  possibility  of  re- 
ferring to  new  variables  all  the  general  differential  for- 
mulas primitively  established  for  others.  This  question 
is  now  resolved  in  the  most  complete  and  the  most  sim- 
ple manner,  as  are  all  those  of  which  the  differential 
calculus  is  composed.  It  is  easy  to  conceive  the  gen- 
eral importance  which  it  must  have  in  any  of  the  appli- 
cations of  the  transcendental  analysis,  the  fundamental 
resources  of  which  it  may  be  considered  as  augmenting, 
by  permitting  us  to  chocse  (in  order  to  form  the  differ- 
ential equations,  in  the  first  place,  with  more  ease)  that 
system  of  independent  variables  which  may  appear  to 
be  the  most  advantageous,  although  it  is  not  to  be  final- 
ly retained.  It  is  thus,  for  example,  that  most  of  the 
principal  questions  of  geometry  are  resolved  much  more 
easily  by  referring  the  lines  and  surfaces  to  rectilinear 
co-ordinates,  and  that  we  may,  nevertheless,  have  occa- 
sion to  express  these  lines,  etc.,  analytically  by  the  aid 
Bipolar  co-ordinates,  or  in  any  other  manner.  We  will 
then  be  able  to  commence  the  differential  solution  of  the 
problem  by  employing  the  rectilinear  system,  but  only 
as  an  intermediate  step,  from  which,  by  the  general  the- 
ory here  referred  to,  we  can  pass  to  the  final  system, 
which  sometimes  could  not  have  been  considered  directly. 


THE   DIFFERENTIAL   CALCULUS.          ^33 

Different  Orders  of  Differentiation.  In  the  logical 
classification  of  the  differential  calculus  which  has  just 
been  given,  some  may  be  inclined  to  suggest  a  serious 
omission,  since  I  have  not  subdivided  each  of  its  four 
essential  parts  according  to  another  general  considera- 
tion, which  seems  at  first  view  very  important ;  namely, 
that  of  the  higher  or  lower  order  of  differentiation.  But 
it  is  easy  to  understand  that  this  distinction  has  no  real 
influence  in  the  differential  calculus,  inasmuch  as  it  does 
not  give  rise  to  any  new  difficulty.  If,  indeed,  the  dif- 
ferential calculus  was  not  rigorously  complete,  that  is, 
if  we  did  not  know  how  to  differentiate  at  will  any  func- 
tion whatever,  the  differentiation  to  the  second  or  higher 
order  of  each  determinate  function  might  engender  spe- 
cial difficulties.  But  the  perfect  universality  of  the  dif- 
ferential calculus  plainly  gives  us  the  assurance  of  being 
able  to  differentiate,  to  any  order  whatever,  all  known 
functions  whatever,  the  question  reducing  itself  to  a  con- 
stantly repeated  differentiation  of  the  first  order.  This 
distinction,  unimportant  as  it  is  for  the  differential  cal- 
culus, acquires,  however,  a  very  great  importance  in  the 
integral  calculus,  on  account  of  the  extreme  imperfection 
of  the  latter. 

Analytical  Applications.  Finally,  though  this  is  not 
the  place  to  consider  the  various  applications  of  the  dif- 
ferential calculus,  yet  an  exception  may  be  made  for 
those  which  consist  in  the  solution  of  questions  which  are 
purely  analytical,  which  ought,  indeed,  to  be  logically 
treated  in  continuation  of  a  system  of  differentiation,  be- 
cause of  the  evident  homogeneity  of  the  considerations 
involved.  These  questions  may  be  reduced  to  three  es- 
sential ones. 


134  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

Firstly,  the  development  into  series  of  functions  of 
one  or  more  variables,  or,  more  generally,  the  transform- 
ation of  functions,  which  constitutes  the  most  beautiful 
and  the  most  important  application  of  the  differential  cal- 
culus to  general  analysis,  and  which  comprises,  besides 
the  fundamental  series  discovered  by  Taylor,  the  remark- 
able series  discovered  by  Maclaurin,  John  Bernouilli,  La- 
grange,  &c. : 

Secondly,  the  general  theory  of  maxima  and  minima 
values  for  any  functions  whatever,  of  one  or  more  varia- 
bles ;  one  of  the  most  interesting  problems  which  anal- 
ysis can  present,  however  elementary  it  may  now  have 
become,  and  to  the  complete  solution  of  which  the  dif- 
ferential calculus  naturally  applies : 

Thirdly,  the  general  determination  of  the  true  value 
of  functions  which  present  themselves  under  an  indeter- 
minate appearance  for  certain  hypotheses  made  on  the 
values  of  the  corresponding  variables ;  which  is  the  least 
extensive  and  the  least  important  of  the  three. 

The  first  question  is  certainly  the  principal  one  in  all 
points  of  view ;  it  is  also  the  most  susceptible  of  receiv- 
ing a  new  extension  hereafter,  especially  by  conceiving, 
in  a  broader  manner  than  has  yet  been  done,  the  em- 
ployment of  the  differential  calculus  in  the  transforma- 
tion of  functions,  on  which  subject  Lagrange  has  left 
some  valuable  hints. 

Having  thus  summarily,  though  perhaps  too  briefly, 
considered  the  chief  points  in  the  differential  calculus,  I 
now  proceed  to  an  equally  rapid  exposition  of  a  syste- 
matic outline  of  the  Integral  Calculus,  properly  so  called, 
that  is,  the  abstract  subject  of  integration. 


THE   INTEGRAL   CALCULUS.  135 


THE    INTEGRAL    CALCULUS. 

Its  Fundamental  Division.  The  fundamental  divi- 
sion of  the  Integral  Calculus  is  founded  on  the  same  prin- 
ciple as  that  of  the  Differential  Calculus,  in  distinguishing 
the  integration  of  explicit  differential  formulas,  and  the 
integration  of  implicit  differentials  or  of  differential  equa- 
tions. The  separation  of  these  two  cases  is  even  much 
more  profound  in  relation  to  integration  than  to  differen- 
tiation. In  the  differential  calculus,  in  fact,  this  dis- 
tinction rests,  as  we  have  seen,  only  on  the  extreme  im- 
perfection of  ordinary  analysis.  But,  on  the  other  hand, 
it  is  easy  to  see  that,  even  though  all  equations  could  be 
algebraically  resolved,  differential  equations  would  none 
the  less  constitute  a  case  of  integration  quite  distinct 
from  that  presented  by  the  explicit  differential  formulas  ; 
for,  limiting  ourselves,  for  the  sake  of  simplicity,  to  the 
first  order,  and  to  a  single  function  y  of  a  single  variable 
x,  if  we  suppose  any  differential  equation  between  x,  y, 

dy  dy 

and  — ,  to  be  resolved  with  reference  to  — ,  the  expres- 
dx  dx 

sion  of  the  derived  function  being  then  generally  found 
to  contain  the  primitive  function  itself,  which  is  the  ob- 
ject of  the  inquiry,  the  question  of  integration  will  not 
have  at  all  changed  its  nature,  and  the  solution  will  not 
really  have  made  any  other  progress  than  that  of  having 
brought  the  proposed  differential  equation  to  be  of  only 
the  first  degree  relatively  to  the  derived  function,  which 
is  in  itself  of  little  importance.  The  differential  would 
not  then  be  determined  in  a  manner  much  less  implicit 
than  before,  as  regards  the  integration,  which  would  con- 
tinue to  present  essentially  the  same  characteristic  diffi- 


136  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

culty.  The  algebraic  resolution  of  equations  could  not 
make  the  case  which  we  are  considering  come  within  the 
simple  integration  of  explicit  differentials,  except  in  the 
special  cases  in  which  the  proposed  differential  equation 
did  not  contain  the  primitive  function  itself,  which  would 

dii 
consequently  permit  us,  by  resolving  it,  to  find  —  in 

dijU 

terms  of  x  only,  and  thus  to  reduce  the  question  to  the 
class  of  quadratures.  Still  greater  difficulties  would  evi- 
dently be  found  in  differential  equations  of  higher  orders, 
or  containing  simultaneously  different  functions  of  sev- 
eral independent  variables. 

The  integration  of  differential  equations  is  then  ne- 
cessarily more  complicated  than  that  of  explicit  differen- 
tials, by  the  elaboration  of  which  last  the  integral  calculus 
has  been  created,  and  upon  which  the  others  have  been 
made  to  depend  as  far  as  it  has  been  possible.  All  the 
various  analytical  methods  which  have  been  proposed  for 
integrating  differential  equations,  whether  it  be  the  sep- 
aration of  the  variables,  the  method  of  multipliers,  &c., 
have  in  fact  for  their  object  to  reduce  these  integrations 
to  those  of  differential  formulas,  the  only  one  which,  by  its 
nature,  can  be  undertaken  directly.  Unfortunately,  im- 
perfect as  is  still  this  necessary  base  of  the  whole  integral 
calculus,  the  art  of  reducing  to  it  the  integration  of  dif- 
ferential equations  is  still  less  advanced. 

Subdivisions :  one  variable  or  several.  Each  of  these 
two  fundamental  branches  of  the  integral  calculus  is  next 
subdivided  into  two  others  (as  in  the  differential  calcu- 
lus, and  for  precisely  analogous  reasons),  according  as  we 
consider  functions  with  a  single  variable,  or  functions 
with  several  independent  variables. 


THE   INTEGRAL    CALCULUS.  ^37 

This  distinction  is,  like  the  preceding  one,  still  more 
important  for  integration  than  for  differentiation.  This 
is  especially  remarkable  in  reference  to  differential  equa- 
tions. Indeed,  those  which  depend  on  several  indepen- 
dent variables  may  evidently  present  this  characteristic 
and  much  more  serious  difficulty,  that  the  desired  func- 
tion may  be  differentially  denned  by  a  simple  relation  be- 
tween its  different  special  derivatives  relative  to  the  dif- 
ferent variables  taken  separately.  Hence  results  the 
most  difficult  and  also  the  most  extensive  branch  of  the 
integral  calculus,  which  is  commonly  named  the  Inte- 
gral Calculus  of  partial  differences,  created  by  D'Alem- 
bert,  and  in  which,  according  to  the  just  appreciation  of 
Lagrange,  geometers  ought  to  have  seen  a  really  new 
calculus,  the  philosophical  character  of  which  has  not  yet 
been  determined  with  sufficient  exactness.  A  very  stri- 
king difference  between  this  case  and  that  of  equations 
with  a  single  independent  variable  consists,  as  has  been 
already  observed,  in  the  arbitrary  functions  which  take 
the  place  of  the  simple  arbitrary  constants,  in  order  to  give 
to  the  corresponding  integrals  all  the  proper  generality. 

It  is  scarcely  necessary  to  say  that  thfs  higher  branch 
of  transcendental  analysis  is  still  entirely  in  its  infancy, 
since,  even  in  the  most  simple  case,  that  of  an  equation 
of  the  first  order  between  the  partial  derivatives  of  a  sin- 
gle function  with  two  independent  variables,  we  are  not 
yet  completely  able  to  reduce  the  integration  to  that  of 
the  ordinary  differential  equations.  The  integration  of 
functions  of  several  variables  is  much  farther  advanced 
in  the  case  (infinitely  more  simple  indeed)  in  which  it 
has  to  do  with  only  explicit  differential  formulas.  We 
can  then,  in  fact,  when  these  formulas  fulfil  the  neces- 


138  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

sary  conditions  of  integrability,  always  reduce  their  in- 
tegration to  quadratures. 

Other  Subdivisions :  different  Orders  of  Differentia- 
tion. A  new  general  distinction,  applicable  as  a  subdi- 
vision to  the  integration  of  explicit  or  implicit  differen- 
tials, with  one  variable  or  several,  is  drawn  from  the  high- 
er or  lower  order  of  the  differentials :  a  distinction  which, 
as  we  have  above  remarked,  does  not  give  rise  to  any 
special  question  in  the  differential  calculus. 

Relatively  to  explicit  differentials,  whether  of  one  va- 
riable or  of  several,  the  necessity  of  distinguishing  their 
different  orders  belongs  only  to  the  extreme  imperfection 
of  the  integral  calculus.  In  fact,  if  we  could  always  in- 
tegrate every  differential  formula  of  the  first  order,  the 
integration  of  a  formula  of  the  second  order,  or  of  any 
other,  would  evidently  not  form  a  new  question,  since,  by 
integrating  it  at  first  'in  the  first  degree,  we  would  arrive 
at  the  differential  expression  of  the  immediately  prece- 
ding order,  from  which,  by  a  suitable  series  of  analogous 
integrations,  we  would  be  certain  of  finally  arriving  at 
the  primitive  function,  the  final  object  of  these  opera- 
tions. But  the  little  knowledge  which  we  possess  on  in- 
tegration of  even  the  first  order  causes  quite  another  state 
of  affairs,  so  that  a  higher  order  of  differentials  produces 
new  difficulties  ;  for,  having  differential  formulas  of  any 
order  above  the  first,  it  may  happen  that  we  may  be  able 
to  integrate  them,  either  once,  or  several  times  in  suc- 
cession, and  that  we  may  still  be  unable  to  go  back  to 
the  primitive  functions,  if  these  preliminary  labours  have 
produced,  for  the  differentials  of  a  lower  order,  expres- 
sions whose  integrals  are  not  known.  This  circumstance 
must  occur  so  much  the  oftener  (the  number  of  known 


THE  INTEGRAL    CALCULUS. 

integrals  being  still  very  small),  seeing  that  these  suc- 
cessive integrals  are  generally  very  different  functions 
from  the  derivatives  which  have  produced  them. 

With  reference  to  implicit  differentials,  the  distinc- 
tion of  orders  is  still  more  important  ;  for,  besides  the 
preceding  reason,  the  influence  of  which  is  evidently 
analogous  in  this  case,  and  is  even  greater,  it  is  easy  to 
perceive  that  the  higher  order  of  the  differential  equa- 
tions necessarily  gives  rise  to  questions  of  a  new  nature. 
In  fact,  even  if  we  could  integrate  every  equation  of  the 
first  order  relating  to  a  single  function,  that  would  not 
be  sufficient  for  obtaining  the  final  integral  of  an  equa- 
tion of  any  order  whatever,  inasmuch  as  every  differential 
equation  is  not  reducible  to  that  of  an  immediately  in- 
ferior order.  Thus,  for  example,  if  we  have  given  any 

/7'T*  fj    1] 

relation  between  x,  y.  —  ,  and  —  —  .  to  determine  a  func- 

z 


dy 

tion  y  of  a  variable  x,  we  shall  not  be  able  to  deduce 
from  it  at  once,  after  effecting  a  first  integration,  the 

dy 

corresponding  differential  relation  between  x,  y,  and  —  , 

££•£ 

from  which,  by  a  second  integration,  we  could  ascend 
to  the  primitive  equations.  This  would  not  necessarily 
take  place,  at  least  without  introducing  new  auxiliary 
functions,  unless  the  proposed  equation  of  the  second  or- 
der did  not  contain  the  required  function  y,  together  with 
its  derivatives.  As  a  general  principle,  differential  equa- 
tions will  have  to  be  regarded  as  presenting  cases  which 
are  more  and  more  implicit,  as  they  are  of  a  higher  or- 
der, and  which  cannot  be  made  to  depend  on  one  another 
except  by  special  methods,  the  investigation  of  which 
consequently  forms  a  new  class  of  questions,  with  re- 


140  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

spect  to  which  we  as  yet  know  scarcely  any  thing,  even 
for  functions  of  a  single  variable.* 

Another  equivalent  distinction.  Still  farther,  when 
we  examine  more  profoundly  this  distinction  of  different 
orders  of  differential  equations,  we  find  that  it  can  be 
always  made  to  come  under  a  final  general  distinction, 
relative  to  differential  equations,  which  remains  to  be 
noticed.  Differential  equations  with  one  or  more  inde- 
pendent variables  may  contain  simply  a  single  function, 
or  (in  a  case  evidently  more  complicated  and  more  im- 
plicit, which  corresponds  to  the  differentiation  of  simul- 
taneous implicit  functions)  we  may  have  to  determine 
at  the  same  time  several  functions  from  the  differential 
equations  in  which  they  are  found  united,  together  with 
their  different  derivatives.  It  is  clear  that  such  a  state 
of  the  question  necessarily  presents  a  new  special  diffi- 
culty, that  of  separating  the  different  functions  desired, 
by  forming  for  each,  from  the  proposed  differential  equa- 
tions, an  isolated  differential  equation  which  does  not 
contain  the  other  functions  or  their  derivatives.  This 
preliminary  labour,  which  is  analogous  to  the  elimina- 
tion of  algebra,  is  evidently  indispensable  before  attempt- 
ing any  direct  integration,  since  we  cannot  undertake 
generally  (except  by  special  artifices  which  are  very 
rarely  applicable)  to  determine  directly  several  distinct 
functions  at  once. 

Now  it  is  easy  to  establish  the  exact  and  necessary 
coincidence  of  this  new  distinction  with  the  preceding 

The  only  important  case  of  this  class  which  has  thus  far  been  com- 
pletely treated  is  the  general  integration  of  linear  equations  of  any  order 
whatever,  with  constant  coefficients.  Even  this  case  finally  depends  on 
the  algebraic  resolution  of  equations  of  a  degree  equal  to  the  order  of  dif- 
ferentiation. 


THE    INTEGRAL   CALCULUS. 

one  respecting  the  order  of  differential  equations.  We 
know,  in  fact,  that  the  general  method  for  isolating  func- 
tions in  simultaneous  differential  equations  consists  es- 
sentially in  forming  differential  equations,  separately  in 
relation  to  each  function,  and  of  an  order  equal  to  the 
sum  of  all  those  of  the  different  proposed  equations. 
This  transformation  can  always  be  effected.  On  the 
other  hand,  every  differential  equation  of  any  order  in 
relation  to  a  single  function  might  evidently  always  be 
reduced  to  the  first  order,  by  introducing  a  suitable  num- 
ber of  auxiliary  differential  equations,  containing  at  the 
same  time  the  different  anterior  derivatives  regarded  as 
new  functions  to  be  determined.  This  method  has,  in- 
deed, sometimes  been  actually  employed  with  success, 
though  it  is  not  the  natural  one. 

Here,  then,  are  two  necessarily  equivalent  orders  of 
conditions  in  the  general  theory  of  differential  equations ; 
the  simultaneousness  of  a  greater  or  smaller  number  of 
functions,  and  the  higher  or  lower  order  of  differentia- 
tion of  a  single  function.  By  augmenting  the  order  of 
the  differential  equations,  we  can  isolate  all  the  func- 
tions ;  and,  by  artificially  multiplying  the  number  of 
the  functions,  we  can  reduce  all  the  equations  to  the 
first  order.  There  is,  consequently,  in  both  cases,  only 
one  and  the  same  difficulty  from  two  different  points  of 
sight.  But,  however  we  may  conceive  it,  this  new  dif- 
ficulty is  none  the  less  real,  and  constitutes  none  the 
less,  by  its  nature,  a  marked  separation  between  the  in- 
tegration of  equations  of  the  first  order  and  that  of  equa- 
tions of  a  higher  order.  I  prefer  to  indicate  the  dis- 
tinction -under  this  last  form  as  being  more  simple,  more 
general,  and  more  logical. 


142  DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

Quadratures.  From  the  different  considerations 
which  have  been  indicated  respecting  the  logical  depend- 
ence of  the  various  principal  parts  of  the  integral  cal- 
culus, we  see  that  the  integration  of  explicit  differential 
formulas  of  the  first  order  and  of  a  single  variable  is  the 
necessary  basis  of  all  other  integrations,  which  we  never 
succeed  in  effecting  but  so  far  as  we  reduce  them  to  this 
elementary  case,  evidently  the  only  one  which,  by  its 
nature,  is  capable  of  being  treated  directly.  This  sim- 
ple fundamental  integration  is  often  designated  by  the 
convenient  expression  of  quadratures,  seeing  that  every 
integral  of  this  kind,  Sf(x)dx,  may,  in  fact,  be  regarded 
as  representing  the  area  of  a  curve,  the  equation  of  which 
in  rectilinear  co-ordinates  would  be  y=f(x\.  Such  a 
class  of  questions  corresponds,  in  the  differential  calculus, 
to  the  elementary  case  of  the  differentiation  of  explicit 
functions  of  a  single  variable.  But  the  integral  ques- 
tion is,  by  its  nature,  very  differently  complicated,  and 
especially  much  more  extensive  than  the  differential 
question.  This  latter  is,  in  fact,  necessarily  reduced,  as 
we  have  seen,  to  the  differentiation  of  the  ten  simple 
functions,  the  elements  of  all  which  are  considered  in 
analysis.  On  the  other  hand,  the  integration  of  com- 
pound functions  does  not  necessarily  follow  from  that  of 
the  simple  functions,  each  combination  of  which  may 
present  special  difficulties  with  respect  to  the  integral 
calculi^.  Hence  results  the  naturally  indefinite  extent, 
and  the  so  varied  complication  of  the  question  of  quadra- 
tures, upon  which,  in  spite  of  all  the  efforts  of  analysts, 
we  still  possess  so  little  complete  knowledge. 

In  decomposing  this  question,  as  is  natural,  according 
to  the  different  forms  which  may  be  assumed  by  the 


THE   INTEGRAL  CALCULUS. 

derivative  function,  we  distinguish  the  case  of  algebraic 
functions  and  that  of  transcendental  functions. 

Integration  of  Transcendental  Functions.  Th<,  tiuly 
analytical  integration  of  transcendental  functions  is  as 
yet  very  little  advanced,  whether  for  exponential,  or  for 
logarithmic,  or  for  circular  functions.  But  a  very  small 
number  of  cases  of  these  three  different  kinds  have  as 
yet  been  treated,  and  those  chosen  from  among  the  sim- 
plest ;  and  still  the  necessary  calculations  are  in  most 
cases  extremely  laborious.  A  circumstance  which  we 
ought  particularly  to  remark  in  its  philosophical  con- 
nection is,  that  the  different  procedures  of  quadrature 
have  no  relation  to  any  general  view  of  integration,  and 
consist  of  simple  artifices  very  incoherent  with  each  other, 
and  very  numerous,  because  of  the  very  limited  extent 
of  each. 

One  of  these  artifices  should,  however,  here  be  no- 
ticed, which,  without  being  really  a  method  of  integra- 
tion, is  nevertheless  remarkable  for  its  generality  ;  it  is 
the  procedure  invented  by  John  Bernouilli,  and  known 
under  the  name  of  integration  by  parts,  by  means  of 
which  every  integral  may  be  reduced  to  another  which 
is  sometimes  found  to  be  more  easy  to  be  obtained. 
This  ingenious  relation  deserves  to  be  noticed  for  anothei 
reason,  as  having  suggested  the  first  idea  of  that  trans- 
formation of  integrals  yet  unknown,  which  has  lately 
received  a  greater  extension,  and  of  which  M.  Fourier 
especially  has  made  so  new  and  important  a  use  in  the 
analytical  questions  produced  by  the  theory  of  heat. 

Integration  of  Algebraic  Functions.  As  to  the  in- 
tegration of  algebraic  functions,  it  is  farther  advanced. 
However,  we  know  scarcely  any  thing  in  relation  to  irra- 


144DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

tional  functions,  the  integrals  of  which  have  been  obtain- 
ed only  in  extremely  limited  cases,  and  particularly  by 
rendering  them  rational.  The  integration  of  rational 
functions  is  thus  far  the  only  theory  of  the  integral  cal- 
culus which  has  admitted  of  being  treated  in  a  truly  com- 
plete manner  ;  in  a  logical  point  of  view,  it  forms,  then, 
its  most  satisfactory  part,  but  perhaps  also  the  least  im- 
portant. It  is  even  essential  to  remark,  in  order  to  have 
a  just  idea  of  the  extreme  imperfection  of  the  integral 
calculus,  that  this  case,  limited  as  it  is,  is  not  entirely 
resolved  except  for  what  properly  concerns  integration 
viewed  in  an  abstract  manner ;  for,  in  the  execution,  the 
theory  finds  its  progress  most  frequently  quite  stopped, 
independently  of  the  complication  of  the  calculations,  by 
the  imperfection  of  ordinary  analysis,  seeing  that  it 
makes  the  integration  finally  depend  upon  the  algebraic 
resolution  of  equations,  which  greatly  limits  its  use. 

To  grasp  in  a  general  manner  the  spirit  of  the  differ- 
ent procedures  which  are  employed  in  quadratures,  we 
must  observe  that,  by  their  nature,  they  can  be  primi- 
tively founded  only  on  the  differentiation  of  the  ten  sim- 
ple functions.  The  results  of  this,  conversely  considered, 
establish  as  many  direct  theorems  of  the  integral  calcu- 
lus, the  only  ones  which  can  be  directly  known.  All  the 
art  of  integration  afterwards  consists,  as  has  been  said 
in  the  beginning  of  this  chapter,  in  reducing  all  the  oth- 
er quadratures,  so  far  as  is  possible,  to  this  small  num- 
ber of  elementary  ones,  which  unhappily  we  are  in  most 
cases  unable  to  effect. 

Singular  Solutions.  In  this  systematic  enumeration 
of  the  various  essential  parts  of  the  integral  calculus,  con- 
sidered in  their  logical  relations,  I  have  designedly  neg- 


THE    INTEGRAL    CALCULUS. 

lected  (in  order  not  to  break  the  chain  of  sequence)  to 
consider  a  very  important  theory,  which  forms  implicitly 
a  portion  of  the  general  theory  of  the  integration  of  dif- 
ferential equations,  but  which  I  ought  here  to  notice  sep- 
arately, as  being,  so  to  speak,  outside  of  the  integral  cal- 
culus, and  being  nevertheless  of  the  greatest  interest,  both 
by  its  logical  perfection  and  by  the  extent  of  its  appli- 
cations. I  refer  to  what  are  called  Singular  Solutions 
of  differential  equations,  called  sometimes,  but  improp- 
erly, particular  solutions,  which  have  been  the  subject 
of  very  remarkable  investigations  by  Euler  and  Laplace, 
and  of  which  Lagrange  especially  has  presented  such  a 
beautiful  and  simple  general  theory.  Clairaut,  who  first 
had  occasion  to  remark  their  existence,  saw  in  them  a 
paradox  of  the  integral  calculus,  since  these  solutions 
have  the  peculiarity  of  satisfying  the  differential  equa- 
tions without  being  comprised  in  the  corresponding  gen- 
eral integrals.  Lagrange  has  since  explained  this  par- 
adox in  the  most  ingenious  and  most  satisfactory  man- 
ner, by  showing  how  such  solutions  are  always  derived 
from  the  general  integral  by  the  variation  of  the  arbi- 
trary constants.  He  was  also  the  first  to  suitably  ap- 
preciate the  importance  of  this  theory,  and  it  is  with 
good  reason  that  he  devoted  to  it  so  full  a  development 
in  his  "Calculus  of  Functions."  In  a  logical  point  of 
view,  this  theory  deserves  all  our  attention  by  the  char- 
acter of  perfect  generality  which  it  admits  of,  since  La- 
grange  has  given  invariable  and  very  simple  procedures 
for  finding  the  singular  solution  of  any  differential  equa- 
tion which  is  susceptible  of  it ;  and,  what  is-  no  less  re- 
markable, these  procedures  require  no  integration,  con- 
sisting only  of  differentiations,  and  are  therefore  always 

K 


146  DIFFERENTIAL  AND  INTEGRAL  CALCULUS. 

applicable.  Differentiation  has  thus  become,  by  a  hap- 
py artifice,  a  means  of  compensating,  in  certain  circum- 
stances, for  the  imperfection  of  the  integral  calculus. 
Indeed,  certain  problems  especially  require,  by  their  na- 
ture, the  knowledge  of  these  singular  solutions ;  such, 
for  example,  in  geometry,  are  all  the  questions  in  which 
a  curve  is  to  be  determined  from  any  property  of  its  tan- 
gent or  its  osculating  circle.  In  all  cases  of  this  kind, 
after  having  expressed  this  property  by  a  differential 
equation,  it  will  be,  in  its  analytical  relations,  the  sin- 
gular equation  which  will  form  the  most  important  ob- 
ject of  the  inquiry,  since  it  alone  will  represent  the  re- 
quired curve;  the  general  integral,  which  thenceforth  it 
becomes  unnecessary  to  know,  designating  only  the  sys- 
tem of  the  tangents,  or  of  the  osculating  circles  of  this 
curve.  We  may  hence  easily  understand  all  the  impor- 
tance of  this  theory,  which  seems  to  me  to  be  not  as  yet 
sufficiently  appreciated  by  most  geometers. 

Definite  Integrals.  Finally,  to  complete  our  review 
of  the  vast  collection  of  analytical  researches  of  which  is 
composed  the  integral  calculus,  properly  so  called,  there 
remains  to  be  mentioned  one  theory,  very  important  in 
all  the  applications  of  the  transcendental  analysis,  which 
I  have  had  to  leave  outside  of  the  system,  as  not  being 
really  destined  for  veritable  integration,  and  proposing,  on 
the  contrary,  to  supply  the  place  of  the  knowledge  of  truly 
analytical  integrals,  which  are  most  generally  unknown. 
I  refer  to  the  determination  of  definite  integrals. 

The  expression,  always  possible,  of  integrals  in  infi- 
nite series,  may  at  first  be  viewed  as  a  happy  general 
means  of  compensating  for  the  extreme  imperfection  of 
the  integral  calculus.  But  the  employment  of  such  se- 


THE  INTEGRAL  CALCULUS.       147 

ries,  because  of  their  complication,  and  of  the  difficulty 
of  discovering  the  law  of  their  terms,  is  commonly  of  only 
moderate  utility  in  the  algebraic  point  of  view,  although 
sometimes  very  essential  relations  have  been  thence  de- 
duced. It  is  particularly  in  the  arithmetical  point  of 
view  that  this  procedure  acquires  a  great  importance,  as 
a  means  of  calculating  what  are  called  definite  integrals, 
that  is,  the  values  of  the  required  functions  for  certain 
determinate  values  of  the  corresponding  variables. 

An  inquiry  of  this  nature  exactly  corresponds,  in  trans- 
cendental analysis,  to  the  numerical  resolution  of  equa- 
tions in  ordinary  analysis.  Being  generally  unable  to 
obtain  the  veritable  integral — named  by  opposition  the 
general  or  indefinite  integral ;  that  is,  the  function  which, 
differentiated,  has  produced  the  proposed  differential  form- 
ula— analysts  have  been  obliged  to  employ  themselves 
in  determining  at  least,  without  knowing  this  function, 
the  particular  numerical  values  which  it  would  take  on 
assigning  certain  designated  values  to  the  variables. 
This  is  evidently  resolving  the  arithmetical  question 
without  having  previously  resolved  the  corresponding  al- 
gebraic one,  which  most  generally  is  the  most  impor- 
tant one.  Such  an  analysis  is,  then,  by  its  nature,  as 
imperfect  as  we  have  seen  the  numerical  resolution  of 
equations  to  be.  It  presents,  like  this  last,  a  vicious 
confusion  of  arithmetical  and  algebraic  considerations, 
whence  result  analogous  inconveniences  both  in  the 
purely  logical  point  of  view  and  in  the  applications. 
We  need  not  here  repeat  the  considerations  suggested  in 
our  third  chapter.  But  it  will  be  understood  that,  un- 
able as  we  almost  always  are  to  obtain  the  true  inte- 
grals, it  is  of  the  highest  importance  to  have  been  able 


148  DIFFERENTIAL   AND   INTEGRAL   CALCULUS. 

to  obtain  this  solution,  incomplete  and  necessarily  insuf- 
ficient as  it  is.  Now  this  has  been  fortunately  attained 
at  the  present  day  for  all  cases,  the  determination  of 
the  value  of  definite  integrals  having  been  reduced  to 
entirely  general  methods,  which  leave  nothing  to  desire, 
in  a  great  number  of  cases,  but  less  complication  in  the 
calculations,  an  object  towards  which  are  at  present  di- 
rected all  the  special  transformations  of  analysts.  Re- 
garding now  this  sort  of  transcendental  arithmetic  as 
perfect,  the  difficulty  in  the  applications  is  essentially 
reduced  to  making  the  proposed  research  depend,  finally, 
on  a  simple  determination  of  definite  integrals,  which 
evidently  cannot  always  be  possible,  whatever  analyti- 
cal skill  may  be  employed  in  effecting  such  a  transfor- 
mation. 

Prospects  of  the  Integral  Calculus.  From  the  con- 
siderations indicated  in  this  chapter,  we  see  that,  while 
the  differential  calculus  constitutes  by  its  nature  a  limited 
and  perfect  system,  to  which  nothing  essential  remains 
to  be  added,  the  integral  calculus,  or  the  simple  system 
of  integration,  presents  necessarily  an  inexhaustible  field 
for  the  activity  of  the  human  mind,  independently  of 
the  indefinite  applications  of  which  the  transcendental 
analysis  is  evidently  susceptible.  The  general  argu- 
ment by  which  I  have  endeavoured,  in  the  second  chap- 
ter, to  make  apparent  the  impossibility  of  ever  discover- 
ing the  algebraic  solution  of  equations  of  any  degree  and 
form  whatsoever,  has  undoubtedly  infinitely  more  force 
with  regard  to  the  search  for  a  single  method  of  integra- 
tion, invariably  applicable  to  all  cases.  "  It  is,"  says 
Lagrange,  "one  of  those  problems  whose  general  solu- 
tion we  cannot  hope  for."  The  more  we  meditate  on 


THE    INTEGRAL   CALCULUS. 

this  subject,  the  more  we  will  be  convinced  that  such  a 
research  is  utterly  chimerical,  as  being  far  above  the  fee- 
ble reach  of  our  intelligence ;  although  the  labours  of 
geometers  must  certainly  augment  hereafter  the  amount 
of  our  knowledge  respecting  integration,  and  thus  create 
methods  of  greater  generality.  The  transcendental  anal- 
ysis is  still  too  near  its  origin — there  is  especially  too 
little  time  since  it  has  been  conceived  in  a  truly  rational 
manner — for  us  now  to  be  able  to  have  a  correct  idea  of 
what  it  will  hereafter  become.  But,  whatever  should  be 
our  legitimate  hopes,  let  us  not  forget  to  consider,  before 
all,  the  limits  which  are  imposed  by  our  intellectual  con- 
stitution, and  which,  though  not  susceptible  of  a  precise 
determination,  have  none  the  less  an  incontestable  reality. 

I  am  induced  to  think  that,  when  geometers  shall  have 
exhausted  the  most  important  applications  of  our  present 
transcendental  analysis,  instead  of  striving  to  impress 
upon  it,  as  now  conceived,  a  chimerical  perfection,  they 
will  rather  create  new  resources  by  changing  the  mode 
of  derivation  of  the  auxiliary  quantities  introduced  in 
order  to  facilitate  the  establishment  of  equations,  and 
the  formation  of  which  might  follow  an  infinity  of  other 
laws  besides  the  very  simple  relation  which  has  been 
chosen,  according  to  the  conception  suggested  in  the  first 
chapter.  The  resources  of  this  nature  appear  to  me  sus- 
ceptible of  a  much  greater  fecundity  than  those  which 
would  consist  of  merely  pushing  farther  our  present  cal- 
culus of  indirect  functions.  It  is  a  suggestion  which  I 
submit  to  the  geometers  who  have  turned  their  thoughts 
towards  the  general  philosophy  of  analysis. 

Finally,  although,  in  the  summary  exposition  which 
was  the  object  of  this  chapter,  I  have  had  to  exhibit  the 


150DIFFERENTIAL  AND  INTEGRAL   CALCULUS. 

condition  of  extreme  imperfection  which  still  belongs  to 
the  integral  calculus,  the  student  would  have  a  false  idea 
of  the  general  resources  of  the  transcendental  analysis  if 
he  gave  that  consideration  too  great  an  importance.  It 
is  with  it,  indeed,  as  with  ordinary  analysis,  in  which  a 
very  small  amount  of  fundamental  knowledge  respecting 
the  resolution  of  equations  has  been  employed  with  an 
immense  degree  of  utility.  Little  advanced  as  geome- 
ters really  are  as  yet  in  the  science  of  integrations,  they 
have  nevertheless  obtained,  from  their  scanty  abstract 
conceptions,  the  solution  of  a  multitude  of  questions  of 
the  first  importance  in  geometry,  in  mechanics,  in  ther- 
mology,  &c.  The  philosophical  explanation  of  this 
double  general  fact  results  from  the  necessarily  prepon- 
derating importance  and  grasp  of  abstract  branches  of 
knowledge,  the  least  of  which  is  naturally  found  to  cor- 
respond to  a  crowd  of  concrete  researches,  man  having 
no  other  resource  for  the  successive  extension  of  his  in- 
tellectual means  than  in  the  consideration  of  ideas  more 
and  more  abstract,  and  still  positive. 

In  order  to  finish  the  complete  exposition  of  the  phil- 
osophical character  of  the  transcendental  analysis,  there 
remains  to  be  considered  a  final  conception,  by  which 
the  immortal  Lagrange  has  rendered  this  analysis  still 
better  adapted  to  facilitate  the  establishment  of  equations 
in  the  most  difficult  problems,  by  considering  a  class  of 
equations  still  more  indirect  than  the  ordinary  differen- 
tial equations.  It  is  the  Calculus,  or,  rather,  the  Method 
of  Variations  ;  the  general  appreciation  of  which  will  be 
our  next  subject. 


CHAPTER    V. 

THE  CALCULUS  OF  VARIATIONS. 

IN  order  to  grasp  with  more  ease  the  philosophical 
character  of  the  Method  of  Variations,  it  will  be  well  to 
begin  by  considering  in  a  summary  manner  the  special 
nature  of  the  problems,  the  general  resolution  of  which 
has  rendered  necessary  the  formation  of  this  hyper-trans- 
cendental analysis.  It  is  still  too  near  its  origin,  and 
its  applications  have  been  too  few,  to  allow  us  to  obtain 
a  sufficiently  clear  general  idea  of  it  from  a  purely  ab- 
stract exposition  of  its  fundamental  theory. 

PROBLEMS    GIVING   RISE    TO   IT. 

The  mathematical  questions  which  have  given  birth 
to  the  Calculus  of  Variations  consist  generally  in  the 
investigation  of  the  maxima  and  minima  of  certain  in- 
determinate integral  formulas,  which  express  the  ana- 
lytical law  of  such  or  such  a  phenomenon  of  geometry 
or  mechanics,  considered  independently  of  any  particular 
subject.  Geometers  for  a  long  time  designated  all  the 
questions  of  this  character  by  the  common  name  of  Iso- 
perimetrical  Problems,  which,  however,  is  really  suita- 
ble to  only  the  smallest  number  of  them. 

Ordinary  Questions  of  Maxima  and  Minima.  In 
the  common  theory  of  maxima  and  minima,  it  is  pro- 
posed to  discover,  with  reference  to  a  given  function  of 
one  or  more  variables,  what  particular  values  must  be 
assigned  to  these  variables,  in  order  that  the  correspond- 


!52         THE   CALCULUS  OF  VARIATIONS. 

ing  value  of  the  proposed  function  may  be  a  maximum 
or  a  minimum  with  respect  to  those  values  which  im- 
mediately precede  and  follow  it ;  that  is,  properly  speak- 
ing, we  seek  to  know  at  what  instant  the  function  ceases 
to  increase  and  commences  to  decrease,  or  reciprocally. 
The  differential  calculus  is  perfectly  sufficient,  as  we 
know,  for  the  general  resolution  of  this  class  of  ques- 
tions, by  showing  that  the  values  of  the  different  varia- 
bles, which  suit  either  the  maximum  or  minimum,  must 
always  reduce  to  zero  the  different  first  derivatives  of 
the  given  function,  taken  separately  with  reference  to 
each  independent  variable,  and  by  indicating,  moreover, 
a  suitable  characteristic  for  distinguishing  the  maximum 
from  the  minimum  ;  consisting,  in  the  case  of  a  function 
of  a  single  variable,  for  example,  in  the  derived  function 
of  the  second  order  taking  a  negative  value  for  the  max- 
imum, and  a  positive  value  for  the  minimum.  Such 
are  the  well-known  fundamental  conditions  belonging  to 
the  greatest  number  of  cases. 

A  new  Class  of  Questions.  The  construction  of  this 
general  theory  having  necessarily  destroyed  the  chief 
interest  which  questions  of  this  kind  had  for  geometers, 
they  almost  immediately  rose  to  the  consideration  of  a 
new  order  of  problems,  at  once  much  more  important  and 
of  much  greater  difficulty — those  of  isoperimeters.  It 
is,  then,  no  longer  the  values  of  the  variables  belonging 
to  the  maximum  or  the  minimum  of  a  given  function 
that  it  is  required  to  determine.  It  is  the  form  of  the 
function  itself  which  is  required  to  be  discovered,  from 
the  condition  of  the  maximum  or  of  the  minimum  of  a 
certain  definite  integral,  merely  indicated,  which  depends 
upon  that  function. 


PROBLEMS  GIVING  RISE   TO   IT. 

Solid  of  least  Resistance.  The  oldest  question  of 
this  nature  is  that  of  the  solid  of  least  resistance,  treat- 
ed by  Newton  in  the  second  book  of  the  Principia,  in 
which  he  determines  what  ought  to  be  the  meridian 
curve  of  a  solid  of  revolution,  in  order  that  the  resistance 
experienced  by  that  body  in  the  direction  of  its  axis 
may  be  th«  least  possible.  But  the  course  pursued  by 
Newton,  from  the  nature  of  his  special  method  of  trans- 
cendental analysis,  had  not  a  character  sufficiently  sim- 
ple, sufficiently  general,  and  especially  sufficiently  ana- 
lytical, to  attract  geometers  to  this  new  order  of  prob- 
lems. To  effect  this,  the  application  of  the  infinitesimal 
method  was  needed  ;  and  this  was  done,  in  1695,  by 
John  Bernoulli,  in  proposing  the  celebrated  problem  of 
the  Brachystochrone. 

This  problem,  which  afterwards  suggested  such  a  long 
series  of  analogous  questions,  consists  in  determining 
the  curve  which  a  heavy  body  must  follow  in  order  to 
descend  from  one  point  to  another  in  the  shortest  possi- 
ble time.  Limiting  the  conditions  to  the  simple  fall 
in  a  vacuum,  the  only  case  which  was  at  first  consid- 
ered, it  is  easily  found  that  the  required  curve  must  be 
a  reversed  cycloid  with  a  horizontal  base,  and  with  its 
origin  at  the  highest  point.  But  the  question  may  be- 
come singularly  complicated,  either  by  taking  into  ac- 
count the  resistance  of  the  medium,  or  the  change  in  the 
intensity  of  gravity. 

Isoperimeters.  Although  this  new  class  of  problems 
was  in  the  first  place  furnished  by  mechanics,  it  is  in 
geometry  that  the  principal  investigations  of  this  char- 
acter were  subsequently  made.  Thus  it  was  proposed 
to  discover  which,  among  all  the  curves  of  the  same  con- 


154    THE  CALCULUS  OF  VARIATIONS. 

tour  traced  between  two  given  points,  is  that  whose  area 
is  a  maximum  or  minimum,  whence  has  come  the  name 
of  Problem  of  Isoperimeters ;  or  it  was  required  that 
the  maximum  or  minimum  should  belong  to  the  surface 
produced  by  the  revolution  of  the  required  curve  about 
an  axis,  or  to  the  corresponding  volume  ;  in  other  cases, 
it  was  the  vertical  height  of  the  center  of  gravity  of  the 
unknown  curve,  or  of  the  surface  and  of  the  volume 
which  it  might  generate,  which  was  to  become  a  maxi- 
mum or  minimum,  &c.  Finally,  these  problems  were 
varied  and  complicated  almost  to  infinity  by  the  Ber- 
nouillis,  by  Taylor,  and  especially  by  Euler,  before  La- 
grange  reduced  their  solution  to  an  abstract  and  en- 
tirely general  method,  the  discovery  of  which  has  put  a 
stop  to  the  enthusiasm  of  geometers  for  such  an  order  of 
"inquiries.  This  is  not  the  place  for  tracing  the  history 
of  this  subject.  I  have  only  enumerated  some  of  the 
simplest  principal  questions,  in  order  to  render  apparent 
the  original  general  object  of  the  method  of  variations. 

Analytical  Nature  of  these  Problems.  We  see  that 
all  these  problems,  considered  in  an  analytical  point  of 
view,  consist,  by  their  nature,  in  determining  what  form 
a  certain  unknown  function  of  one  or  more  variables 
ought  to  have,  in  order  that  such  or  such  an  integral, 
dependent  upon  that  function,  shall  have,  within  assign- 
ed limits,  a  value  which  is  a  maximum  or  a  minimum 
with  respect  to  all  those  which  it  would  take  if  the  re- 
quired function  had  any  other  form  whatever. 

Thus,  for  example,  in  the  problem  of  the  brachysto- 
chrone,  it  is  well  known  that  if  y=f(z),  x=<j>(z),  are  the 
rectilinear  equations  of  the  required  curve,  supposing 
the  axes  of  x  and  of  y  to  be  horizontal,  and  the  axis  of 


FORMER   METHODS.  155 

z  to  be  vertical,  the  time  of  the  fall  of  a  heavy  body  in 
that  curve  from  the  point  whose  ordinate  is  zv  to  that 
whose  ordinate  is  s^  is  expressed  in  general  terms  by 
the  definite  integral 

*))'+(*'(*)  )V^ 


It  is,  then,  necessary  to  find  what  the  two  unknown 
functions  /  and  </>  must  be,  in  order  that  this  integral 
may  be  a  minimum. 

In  the  same  way,  to  demand  what  is  the  curve  among 
all  plane  isoperimetrical  curves,  which' includes  the  great- 
est area,  is  the  same  thing  as  to  propose  to  find,  among 
all  the  functions  f(x)  which  can  give  a  certain  constant 
value  to  the  integral 


that  one  which  renders  the  integral  f  f(x\dx,  taken  be- 
tween the  same  limits,  a  maximum.  It  is  evidently  al- 
ways so  in  other  questions  of  this  class. 

Methods  of  the  older  Geometers.  In  the  solutions 
which  geometers  before  Lagrange  gave  of  these  prob- 
lems, they  proposed,  in  substance,  to  reduce  them  to  the 
ordinary  theory  of  maxima  and  minima.  But  the  means 
employed  to  effect  this  transformation  consisted  in  spe- 
cial simple  artifices  peculiar  to  each  case,  and  the  dis- 
covery of  which  did  not  admit  of  invariable  and  certain 
rules,  so  that  every  really  new  question  constantly  re- 
produced analogous  difficulties,  without  the  solutions  pre- 
viously obtained  being  really  of  any  essential  aid,  other- 
wise than  by  their  discipline  and  training  of  the  mind. 
In  a  word,  this  branch  of  mathematics  presented,  then, 
the  necessary  imperfection  which  always  exists  when  the 
part  common  to  all  questions  of  the  same  class  has  not 


!56         THE   CALCULUS   OF   VARIATIONS. 

yet  been  distinctly  grasped  in  order  to  be  treated  in  an 
abstract  and  thenceforth  general  manner. 

METHOD   OF    LAGRANGE. 

Lagrange,  in  endeavouring  to  bring  all  the  different 
problems  of  isoperimeters  to  depend  upon  a  common  anal- 
ysis, organized  into  a  distinct  calculus,  was  led  to  con- 
ceive a  new  kind  of  differentiation,  to  which  he  has  ap- 
plied the  characteristic  8,  reserving  the  characteristic  d 
for  the  common  differentials.  These  differentials  of  a 
new  species,  which  he  has  designated  under  the  name  of 
Variations,  consist  of  the  infinitely  small  increments 
which  the  integrals  receive,  not  by  virtue  of  analogous 
increments  on  the  part  of  the  corresponding  variables,  as 
in  the  ordinary  transcendental  analysis,  but  by  supposing 
that  the  form  of  the  function  placed  under  the  sign  of 
integration  undergoes  an  infinitely  small  change.  This 
distinction  is  easily  conceived  with  reference  to  curves, 
in  which  we  see  the  ordinate,  or  any  other  variable  of 
the  curve,  admit  of  two  sorts  of  differentials,  evidently 
very  different,  according  as  we  pass  from  one  point  to  an- 
other infinitely  near  it  on  the  same  curve,  or  to  the  cor- 
responding point  of  the  infinitely  near  curve  produced  by 
a  certain  determinate  modification  of  the  first  curved  It 
is  moreover  clear,  that  the  relative  variations  of  differ- 
ent magnitudes  connected  with  each  other  by  any  laws 
whatever  are  calculated,  all  but  the  characteristic,  almost 
exactly  in  the  same  manner  as  the  differentials.  Finally, 

*  Leibnitz  had  already  considered  the  comparison  of  one  curve  with  an 
other  infinitely  near  to  it,  calling  it  "  Differential™  de  curva  in  curvam." 
But  this  comparison  had  no  analogy  with  the  conception  of  Lagrange,  the 
curves  of  Leibnitz  being  embraced  in  the  same  general  equation,  from  which 
they  were  deduced  by  the  simple  change  of  an  arbitrary  constant. 


METHOD    OF    LAGRANGE.  ^57 

from  the  general  notion  of  variations  are  in  like  manner 
deduced  the  fundamental  principles  of  the  algorithm 
proper  to  this  method,  consisting  simply  in  the  evidently 
permissible  liberty  of  transposing  at  will  the  characteris- 
tics specially  appropriated  to  variations,  before  or  after 
those  which  correspond  to  the  ordinary  differentials. 

This  abstract  conception  having  been  once  formed,  La- 
grange  was  able  to  reduce  with  ease,  and  in  the  most 
general  manner,  all  the  problems  of  Isoperimeters  to  the 
simple  ordinary  theory  of  maxima  and  minima.  To  ob- 
tain a  clear  idea  of  this  great  and  happy  transformation, 
we  must  previously  consider  an  essential  distinction  which 
arises  in  the  different  questions  of  isoperimeters. 

Two  Classes  of  Questions.  These  investigations 
must,  in  fact,  be  divided  into  two  general  classes,  ac- 
cording as  the  maxima  and  minima  demanded  are  abso- 
lute or  relative,  to  employ  the  abridged  expressions  of 
geometers. 

Questions  of  the  first  Class.  The  first  case  is  that 
in  which  the  indeterminate  definite  integrals,  the  maxi- 
mum or  minimum  of  which  is  sought,  are  not  subjected, 
by  the  nature  of  the  problem,  to  any  condition ;  as  hap- 
pens, for  example,  in  the  problem  of  the  br achy stochr one, 
in  which  the  choice  is  to  be  made  between  all  imagina- 
ble curves.  The  second  case  takes  place  when,  on  the 
contrary,  the  variable  integrals  can  vary  only  according 
to  certain  conditions,  which  usually  consist  in  other  defi- 
nite integrals  (which  depend,  in  like  manner,  upon  the 
required  functions)  always  retaining  the  same  given  val- 
ue ;  as,  for  example,  in  all  the  geometrical  questions  re- 
lating to  real  isoperimetrical  figures,  and  in  which,  by 
the  nature  of  the  problem,  the  integral  relating  to  the 


158    THE  CALCULUS  OF  VARIATIONS. 

length  of  the  curve,  or  to  the  area  of  the  surface,  must 
remain  constant  during  the  variation  of  that  integral 
which  is  the  object  of  the  proposed  investigation. 

The  Calculus  of  Variations  gives  immediately  the 
general  solution  of  questions  of  the  former  class  ;  for  it 
evidently  follows,  from  the  ordinary  theory  of  maxima 
and  minima,  that  the  required  relation  must  reduce  to 
zero  the  variation  of  the  proposed  integral  with  reference 
to  each  independent  variable  ;  which  gives  the  condition 
common  to  both  the  maximum  and  the  minimum  :  and, 
as  a  characteristic  for  distinguishing  the  one  from  the 
other,  that  the  variation  of  the  second  order  of  the  same 
integral  must  be  negative  for  the  maximum  and  positive 
for  the  minimum.  Thus,  for  example,  in  the  problem 
of  the  brachystochrone,  we  will  have,  in  order  to  deter- 
mine the  nature  of  the  curve  sought,  the  equation-  of 
condition 


2gz 

which,  being  decomposed  into  two,  with  respect  to  the 
two  unknown  functions  /  and  0,  which  are  independent 
of  each  other,  will  completely  express  the  analytical 
definition  of  the  required  curve.  The  only  difficulty 
peculiar  to  this  new  analysis  consists  in  the  elimination 
of  the  characteristic  S,  for  which  the  calculus  of  varia- 
tions furnishes  invariable  and  complete  rules,  founded,  in 
general,  on  the  method  of  "  integration  by  parts,"  from 
which  Lagrange  has  thus  derived  immense  advantage. 
The  constant  object  of  this  first  analytical  elaboration 
(which  this  is  not  the  place  for  treating  in  detail)  is  to 
arrive  at  real  differential  equations,  which  can  always 
be  done  ;  and  thereby  the  question  comes  under  the  or- 


METHOD   OF  LAGRANGE. 

dinary  transcendental  analysis,  which  furnishes  the  solu- 
tion, at  least  so  far  as  to  reduce  it  to  pure  algebra  if 
the  integration  can  be  effected.  The  general  object  of 
the  method  of  variations  is  to  effect  this  transformation, 
for  which  Lagrange  has  established  rules,  which  are  sim- 
ple, invariable,  and  certain  of  success. 

Equations  of  Limits.  Among  the  greatest  special 
advantages  of  the  method  of  variations,  compared  with 
the  previous  isolated  solutions  of  isoperimetrical  prob- 
lems, is  the  important  consideration  of  what  Lagrange 
calls  Equations  of  Limits,  which  were  entirely  neglect- 
ed before  him,  though  without  them  the  greater  part  of 
the  particular  solutions  remained  necessarily  incomplete. 
When  the  limits  of  the  proposed  integrals  are  to  be  fix- 
ed, their  variations  being  zero,  there  is  no  occasion  for 
noticing  them.  But  it  is  no  longer  so  when  these  limits, 
instead  of  being  rigorously  invariable,  are  only  subjected 
to  certain  conditions ;  as,  for  example,  if  the  two  points 
between  which  the  required  curve  is  to  be  traced  are 
not  fixed,  and  have  only  to  remain  upon  given  lines  or 
surfaces.  Then  it  is  necessary  to  pay  attention  to  the 
variation  of  their  co-ordinates,  and  to  establish  between 
them  the  relations  which  correspond  to  the  equations  of 
these  lines  or  of  these  surfaces. 

A  more  general  consideration.  This  essential  con- 
sideration is  only  the  final  complement  of  a  more  gen- 
eral and  more  important  consideration  relative  to  the 
variations  of  different  independent  variables.  If  these 
variables  are  really  independent  of  one  another,  as  when 
we  compare  together  all  the  imaginable  curves  suscepti- 
ble of  being  traced  between  two  points,  it  will  be  the 
same  with  their  variations,  and,  consequently,  the  terms 


IQQ    THE  CALCULUS  OF  VARIATIONS. 

relating  to  each  of  these  variations  will  have  to  be  sep- 
arately equal  to  zero  in  the  general  equation  which  ex- 
presses the  maximum  or  the  minimum.  But  if,  on  the 
contrary,  we  suppose  the  variables  to  be  subjected  to  any 
fixed  conditions,  it  will  be  necessary  to  take  notice  of  the 
resulting  relation  between  their  variations,  so  that  the 
number  of  the  equations  into  which  this  general  equa- 
tion is  then  decomposed  is  always  equal  to  only  the 
number  of  the  variables  which  remain  truly  independ- 
ent. It  is  thus,  for  example,  that  instead  of  seeking 
for  the  shortest  path  between  any  two  points,  in  choosing 
it  from  among  all  possible  ones,  it  may  be  proposed  to 
find  only  what  is  the  shortest  among  all  those  which 
may  be  taken  on  any  given  surface  ;  a  question  the  gen- 
eral solution  of  which  forms  certainly  one  of  the  most 
beautiful  applications  of  the  method  of  variations. 

Questions  of  the  second  Class.  Problems  in  which 
such  modifying  conditions  are  considered  approach  very 
nearly,  in  their  nature,  to  the  second  general  class  of 
applications  of  the  method  of  variations,  characterized 
above  as  consisting  in  the  investigation  of  relative  max- 
ima and  minima.  There  is,  however,  this  essential  dif- 
ference between  the  two  cases,  that  in  this  last  the 
modification  is  expressed  by  an  integral  which  depends 
upon  the  function  sought,  while  in  the  other  it  is  desig- 
nated by  a  finite  equation  which  is  immediately  given. 
It  is  hence  apparent  that  the  investigation  of  relative 
maxima  and  minima  is  constantly  and  necessarily  more 
complicated  than  that  of  absolute  maxima  and  "minima. 
Luckily,  a  very  important  general  theory,  discovered  by 
the  genius  of  the  great  Euler  befo/e  the  invention  of 
the  Calculus  of  Variations,  gives  a  uniform  and  very 


METHOD   OF   LA  GRANGE.  161 

simple  means  of  making  one  of  these  two  classes  of 
questions  dependent  on  the  other.  It  consists  in  this, 
that  if  we  add  to  the  integral  which  is  to  be  a  maximum 
or  a  minimum,  a  constant  and  indeterminate  multiple 
of  that  one  which,  by  the  nature  of  the  problem,  is  to 
remain  constant,  it  will  be  sufficient  to  seek,  by  the  gen- 
eral method  of  Lagrange  above  indicated,  the  absolute 
maximum  or  minimum  of  this  whole  expression.  It 
can  be  easily  conceived,  indeed,  that  the  part  of  the  com- 
plete variation  which  would  proceed  from  the  last  in- 
tegral must  be  equal  to  zero  (because  of  the  constant 
character  of  this  last)  as  well  as  the  portion  due  to  the 
first  integral,  which  disappears  by  virtue  of  the  maxi- 
mum or  minimum  state.  These  two  conditions  evi- 
dently unite  to  produce,  in  that  respect,  effects  exactly 
alike. 

Such  is  a  sketch  of  the  general  manner  in  which  the 
method  of  variation  is  applied  to  all  the  different  ques- 
tions which  compose  what  is  called  the  Theory  of  Is  ope  - 
rimeters.  It  will  undoubtedly  have  been  remarked  in 
this  summary  exposition  how  much  use  has  been  made 
in  this  new  analysis  of  the  second  fundamental  property 
of  the  transcendental  analysis  noticed  in  the  third  chap- 
ter, namely,  the  generality  of  the  infinitesimal  expres- 
sions for  the  representation  of  the  same  geometrical  or 
mechanical  phenomenon,  in  whatever  body  it  may  be 
considered.  Upon  this  generality,  indeed,  are  founded, 
by  their  nature,  all  the  solutions  due  to  the  method  of 
variations.  If  a  single  formula  could  not  express  the 
length  or  the  area  of  any  curve  whatever ;  if  another 
fixed  formula  could  not  designate  the  time  of  the  fall  of 
a  heavy  body,  according  to  whatever  .ine  it  may  de- 

L 


J62    THE  CALCULUS  OF  VARIATIONS. 

scend,  &c.,  how  would  it  have  been  possible  to  resolve 
questions  which  unavoidably  require,  by  their  nature,  the 
simultaneous  consideration  of  all  the  cases  which  can  be 
determined  in  each  phenomenon  by  the  different  subjects 
which  exhibit  it. 

Other  Applications  of  this  Method.  Notwithstand- 
ing the  extreme  importance  of  the  theory  of  isoperime- 
ters,  and  though  the  method  of  variations  had  at  first  no 
other  object  than  the  logical  and  general  solution  of  this 
order  of  problems,  we  should  still  have  but  an  incom- 
plete idea  of  this  beautiful  analysis  if  we  limited  its 
destination  to  this.  In  fact,  the  abstract  conception  of 
two  distinct  natures  of  differentiation  is  evidently  appli- 
cable not  only  to  the  cases  for  which  it  was  created,  but 
also  to  all  those  which  present,  for  any  reason  whatever, 
two  different  manners  of  making  the  same  magnitudes 
vary.  It  is  in  this  way  that  Lagrange  himself  has  made, 
in  his  " Mecanique  Analytique"  an  extensive  and  im- 
portant application  of  his  calculus  of  variations,  by  em- 
ploying it  to  distinguish  the  two  sorts  of  changes  which 
are  naturally  presented  by  the  questions  of  rational  me- 
chanics for  the  different  points  which  are  considered,  ac- 
cording as  we  compare  the  successive  positions  which 
are  occupied,  in  virtue  of  its  motion,  by  the  same  point 
of  each  body  in  two  consecutive  instants,  or  as  we  pass 
from  one  point  of  the  body  to  another  in  the  same  instant. 
One  of  these  comparisons  produces  ordinary  differentials; 
the  other  gives  rise  to  variations,  which,  there  as  every 
where,  are  only  differentials  taken  under  a  new  point  of 
view.  Such  is  the  general  acceptation  in  which  we 
should  conceive  the  Calculus  of  Variations,  in  order  suit- 
ably to  appreciate  the  importance  of  this  admirable  log- 


RELATIONS  TO  THE  ORDINARY  CALCULUS.  163 

ical  instrument,  the  most  powerful  that  the  human  mind 
has  as  yet  constructed. 

The  method  of  variations  being  only  an  immense  ex- 
tension of  the  general  transcendental  analysis,  I  have  no 
need  of  proving  specially  that  it  is  susceptible  of  being 
considered  under  the  different  fundamental  points  of  view 
which  the  calculus  of  indirect  functions,  considered  as  a 
whole,  admits  of.  Lagrange  invented  the  Calculus  of 
Variations  in  accordance  with  the  infinitesimal  concep- 
tion, and,  indeed,  long  before  he  undertook  the  general  re- 
construction of  the  transcendental  analysis.  When  he 
had  executed  this  important  reformation,  he  easily  showed 
how  it  could  also  be  applied  to  the  Calculus  of  Varia- 
tions, which  he  expounded  with  all  the  proper  develop- 
ment, according  to  his  theory  of  derivative  functions. 
But  the  more  that  the  use  of  the  method  of  variations  is 
difficult  of  comprehension,  because  of  the  higher  degree 
of  abstraction  of  the  ideas  considered,  the  more  necessary 
is  it,  in  its  application,  to  economize  the  exertions  of  the 
mind,  by  adopting  the  most  direct  and  rapid  analytical 
conception,  namely,  that  of  Leibnitz.  Accordingly,  La- 
grange  himself  has  constantly  preferred  it  in  the  impor- 
tant use  which  he  has  made  of  the  Calculus  of  Varia- 
tions in  his  "  Analytical  Mechanics."  In  fact,  there  does 
not  exist  the  least  hesitation  in  this  respect  among  ge- 
ometers. 

ITS  RELATIONS  TO  THE  ORDINARY  CALCULUS. 

In  order  to  make  as  clear  as  possible  the  philosophical 
character  of  the  Calculus  of  Variations,  I  think  that  I 
should,  in  conclusion,  briefly  indicate  a  consideration 
which  seems  to  me  important,  and  by  which  I  can  ap- 


164    THE  CALCULUS  OF  VARIATIONS. 

proach  it  to  the  ordinary  transcendental  analysis  in  a 
higher  degree  than  Lagrange  seems  to  me  to  have  done.* 
We  noticed  in  the  preceding  chapter  the  formation  of 
the  calculus  of  partial  differences,  created  by  D'Alem- 
bert,  as  having  introduced  into  the  transcendental  analy- 
sis a  new  elementary  idea ;  the  notion  of  two  kinds  of 
increments,  distinct  and  independent  of  one  another, 
which  a  function  of  two  variables  may  receive  by  virtue 
of  the  change  of  each  variable  separately.  It  is  thus 
that  the  vertical  ordinate  of  a  surface,  or  any  other  mag- 
nitude which  is  referred  to  it,  varies  in  two  manners 
which  are  quite  distinct,  and  which  may  follow  the  most 
different  laws,  according  as  we  increase  either  the  one 
or  the  other  of  the  two  horizontal  co-ordinates.  Now 
such  a  consideration  seems  to  me  very  nearly  allied,  by 
its  nature,  to  that  which  serves  as  the  general  basis  of 
the  method  of  variations.  This  last,  indeed,  has  in  real- 
ity done  nothing  but  transfer  to  the  independent  varia- 
bles themselves  the  peculiar  conception  which  had  been 
already  adopted  for  the  functions  of  these  variables ;  a 
modification  which  has  remarkably  enlarged  its  use.  I 
think,  therefore,  that  so  far  as  regards  merely  the  funda- 
mental conceptions,  we  may  consider  the  calculus  created 
by  D'Alembert  as  having  established  a  natural  and  ne- 
cessary transition  between  the  ordinary  infinitesimal  cal- 
culus and  the  calculus  of  variations ;  such  a  derivation 
of  which  seems  to  be  adapted  to  make  the  general  notion 
more  clear  and  simple. 


*  I  propose  hereafter  to  develop  this  new  consideration,  in  a  special  work 
upon  the  Calculut  of  Variations,  intended  to  present  this  hyper-transcen- 
dental analysis  in  a  new  point  of  view,  which  I  think  adapted  to  extend  its 
general  range. 


RELATIONS   TO  THE  ORDINARY  CALCULUS.  165 

According  to  the  different  considerations  indicated  in 
this  chapter,  the  method  of  variations  presents  itself  as . 
the  highest  degree  of  perfection  which  the  analysis  of  in- 
direct functions  has  yet  attained.  In  its  primitive  state, 
this  last  analysis  presented  itself  as  a  powerful  general 
means  of  facilitating  the  mathematical  study  of  natural 
phenomena,  by  introducing,  for  the  expression  of  their 
laws,  the  consideration  of  auxiliary  magnitudes,  chosen 
in  such  a  manner  that  their  relations  are  necessarily  more 
simple  and  more  easy  to  obtain  than  those  of  the  direct 
magnitudes.  But  the  formation  of  these  differential 
equations  was  not  supposed  to  admit  of  any  general  and 
abstract  rules.  Now  the  Analysis  of  Variations,  con- 
sidered in  the  most  philosophical  point  of  view,  may  be 
regarded  as  essentially  destined,  by  its  nature,  to  bring 
within  the  reach  of  the  calculus  the  actual  establishment 
of  the  differential  equations ;  for,  in  a  great  number  of 
important  and  difficult  questions,  such  is  the  general  ef- 
fect of  the  varied  equations,  which,  still  more  indirect 
than  the  simple  differential  equations  with  respect  to  the 
special  objects  of  the  investigation,  are  also  much  more 
easy  to  form,  and  from  which  we  may  then,  by  invaria- 
ble and  complete  analytical  methods,  the  object  of  which 
is  to  eliminate  the  new  order  of  auxiliary  infinitesimals 
which  have  been  introduced,  deduce  those  ordinary  differ- 
ential equations  which  it  would  often  have  been  impos- 
sible to  establish  directly.  The  method  of  variations 
forms,  then,  the  most  sublime  paxt  of  that  vast  system 
of  mathematical  analysis,  which,  setting  out  from  the 
most  simple  elements  of  algebra,  organizes,  by  an  unin- 
terrupted succession  of  ideas,  general  methods  more  and 
more  powerful,  for  the  study  of  natural  philosophy,  and 


166    THE  CALCULUS  OF  VARIATIONS. 

which,  in  its  whole,  presents  the  most  incomparably  im- 
posing and  unequivocal  monument  of  the  power  of  the 
human  intellect. 

We  must,  however,  also  admit  that  the  conceptions 
which  are  habitually  considered  in  the  method  of  varia- 
tions being,  by  their  nature,  more  indirect,  more  gen- 
eral, and  especially  more  abstract  than  all  others,  the 
employment  of  such  a  method  exacts  necessarily  and 
continuously  the  highest  known  degree  of  intellectual 
exertion,  in  order  never  to  lose  sight  of  the  precise  ob- 
ject of  the  investigation,  in  following  reasonings  which 
offer  to  the  mind  such  uncertain  resting-places,  and  in 
which  signs  are  of  scarcely  any  assistance.  We  must 
undoubtedly  attribute  in  a  great  degree  to  this  difficulty 
the  little  real  use  which  geometers,  with  the  exception 
of  Lagrange,  have  as  yet  made  of  such  an  admirable 
conception. 


CHAPTER    VI. 

THE  CALCULUS  OF  FINITE  DIFFERENCES. 

THE  different  fundamental  considerations  indicated  in 
the  five  preceding  chapters  constitute,  in  reality,  all  the 
essential  bases  of  a  complete  exposition  of  mathematical 
analysis,  regarded  in  the  philosophical  point  of  view. 
Nevertheless,  in  order  not  to  neglect  any  truly  impor- 
tant general  conception  relating  to  this  analysis,  I  think 
that  I  should  here  very  summarily  explain  the  veritable 
character  of  a  kind  of  calculus  which  is  very  extended, 
and  which,  though  at  bottom  it  really  belongs  to  ordina- 
ry analysis,  is  still  regarded  as  being  of  an  essentially 
distinct  nature.  I  refer  to  the  Calculus  of  Finite  Dif- 
ferences, which  will  be  the  special  subject  of  this  chapter. 

Its  general  Character.  This  calculus,  created  by 
Taylor,  in  his  celebrated  work  entitled  Methodus  Incre- 
mentorum,  consists  essentially  in  the  consideration  of  the 
finite  increments  which  functions  receive  as  a  conse- 
quence of  analogous  increments  on  the  part  of  the  cor- 
responding variables.  These  increments  or  differences, 
which  take  the  characteristic  A,  to  distinguish  them  from 
differentials,  or  infinitely  small  increments,  may  be  in 
their  turn  regarded  as  new  functions,  and  become  the 
subject  of  a  second  similar  consideration,  and  so  on ;  from 
which  results  the  notion  of  differences  of  various  suc- 
cessive orders,  analogous,  at  least  in  appearance,  to  the 
consecutive  orders  of  differentials.  Such  a  calculus  evi- 


J63  THE   CALCULUS  OF  FINITE  DIFFERENCES. 

dently  presents,  like  the  calculus  of  indirect  functions, 
two  general  classes  of  questions  : 

1°.  To  determine  the  successive  differences  of  all  the 
various  analytical  functions  of  one  or  more  variables,  as 
the  result  of  a  definite  manner  of  increase  of  the  inde- 
pendent variables,  which  are  generally  supposed  to  aug- 
ment in  arithmetical  progression : 

2°.  Reciprocally,  to  start  from  these  differences,  or, 
more  generally,  from  any  equations  established  between 
them,  and  go  back  to  the  primitive  functions  themselves, 
or  to  their  corresponding  relations. 

Hence  follows  the  decomposition  of  this  calculus  into 
two  distinct  ones,  to  which  are  usually  given  the  names 
of  the  Direct,  and  the  Inverse  Calculus  of  Finite  Differ- 
ences, the  latter  being  also  sometimes  called  the  Integral 
Calculus  of  Finite  Differences.  Each  of  these  would, 
also,  evidently  admit  of  a  logical  distribution  similar  to 
that  given  in  the  fourth  chapter  for  the  differential  and 
the  integral  calculus. 

Its  true  Nature.  There  is  no  doubt  that  Taylor 
thought  that  by  such  a  conception  he  had  founded  a  cal- 
culus of  an  entirely  new  nature,  absolutely  distinct  from 
ordinary  analysis,  and  more  general  than  the  calculus  of 
Leibnitz,  although  resting  on  an  analogous  consideration. 
It  is  in  this  way,  also,  that  almost  all  geometers  have 
viewed  the  analysis  of  Taylor  ;  but  Lagrange,  with  his 
usual  profundity,  clearly  perceived  that  these  properties 
belonged  much  more  to  the  forms  and  to  the  notations 
employed  by  Taylor  than  to  the  substance  of  his  theory. 
In  fact,  that  which  constitutes  the  peculiar  character  of 
the  analysis  of  Leibnitz,  and  makes  of  it  a  truly  distinct 
and  superior  calculus,  is  the  circumstance  that  the  de- 


ITS   TRUE   NATURE.  169 

rived  functions  are  in  general  of  an  entirely  different  na- 
ture from  the  primitive  functions,  so  that  they  may  give 
rise  to  more  simple  and  more  easily  formed  relations ; 
whence  result  the  admirable  fundamental  properties  of 
the  transcendental  analysis,  which  have  been  already  ex- 
plained. But  it  is  not  so  with  the  differences  consider- 
ed by  Taylor  ;  for  these  differences  are,  by  their  nature, 
functions  essentially  similar  to  those  which  have  pro- 
duced them,  a  circumstance  which  renders  them  un- 
suitable to  facilitate  the  establishment  of  equations,  and 
prevents  their  leading  to  more  general  relations.  Every 
equation  of  finite  differences  is  truly,  at  bottom,  an  equa- 
tion directly  relating  to  the  very  magnitudes  whose  suc- 
cessive states  are  compared.  The  scaffolding  of  new 
signs,  which  produce  an  illusion  respecting  the  true  char- 
acter of  these  equations,  disguises  it,  however,  in  a  very 
imperfect  manner,  since  it  could  always  be  easily  made 
apparent  by  replacing  the  differences  by  the  equivalent 
combinations  of  the  primitive  magnitudes,  of  which  they 
are  really  only  the  abridged  designations.  Thus  the  cal- 
culus of  Taylor  never  has  offered,  and  never  can  offer,  in 
any  question  of  geometry  or  of  mechanics,  that  power- 
ful general  aid  which  we  have  seen  to  result  necessarily 
from  the  analysis  of  Leibnitz.  Lagrange  has,  moreover, 
very  clearly  proven  that  the  pretended  analogy  observed 
between  the  calculus  of  differences  and  the  infinitesimal 
calculus  was  radically  vicious,  in  this  way,  that  the  for- 
mulas belonging  to  the  former  calculus  can  never  fur- 
nish, as  particular  cases,  those  which  belong  to  the  lat- 
ter, the  nature  of  which  is  essentially  distinct. 

From  these  considerations  I  am  led  to  think  that  the 
calculus  of  finite  differences  is,  in  general,  improperly 


170THE   CALCULUS   OF   FINITE    DIFFERENCES- 

classed  with  the  transcendental  analysis  proper,  that  is. 
with  the  calculus  of  indirect  functions.  I  consider  it,  on 
the  contrary,  in  accordance  with  the  views  of  Lagrange, 
to  be  only  a  very  extensive  and  very  important  branch 
of  ordinary  analysis,  that  is  to  say,  of  that  which  I 
have  named  the  calculus  of  direct  functions,  the  equa- 
tions which  it  considers  being  always,  in  spite  of  the 
notation,  simple  direct  equations. 

GENERAL    THEORY   OF    SERIES. 

To  sum  up  as  briefly  as  possible  the  preceding  ex- 
planation, the  calculus  of  Taylor  ought  to  be  regarded 
as  having  constantly  for  its  true  object  the  general  the- 
ory of  Series,  the  most  simple  cases  of  which  had  alone 
been  considered  before  that  illustrious  geometer.  I 
ought,  properly,  to  have  mentioned  this  important  the- 
ory in  treating,  in  the  second  chapter,  of  Algebra  proper, 
of  which  it  is  such  an  extensive  branch.  But,  in  order 
to  avoid  a  double  reference  to  it,  I  have  preferred  to  no- 
tice it  only  in  the  consideration  of  the  calculus  of  finite 
differences,  which,  reduced  to  its  most  simple  general 
expression,  is  nothing  but  a  complete  logical  study  of 
questions  relating  to  series. 

Every  Series,  or  succession  of  numbers  deduced  from 
one  another  according  to  any  constant  law,  necessarily 
gives  rise  to  these  two  fundamental  questions : 

1°.  The  law  of  the  series  being  supposed  known,  to 
find  the  expression  for  its  general  term,  so  as  to  be  able 
to  calculate  immediately  any  term  whatever  without  be- 
ing obliged  to  form  successively  all  the  preceding  terms  : 

2°.  In  the  same  circumstances,  to  determine  the  sum 
of  any  number  of  terms  of  the  series  by  means  of  their 


THEORY   OF    SERIES.  171 

places,  so  that  it  can  be  known  without  the  necessity 
of  continually  adding  these  terms  together. 

These  two  fundamental  questions  being  considered  to 
be  resolved,  it  may  be  proposed,  reciprocally,  to  find  the 
law  of  a  series  from  the  form  of  its  general  term,  or  the 
expression  of  the  sum.  Each  of  these  different  problems 
has  so  much  the  more  extent  and  difficulty,  as  there 
can  be  conceived  a  greater  number  of  different  laws  for 
the  series,  according  to  the  number  of  preceding  terms 
on  which  each  term  directly  depends,  and  according  to 
the  function  which  expresses  that  dependence.  We  may 
even  consider  series  with  several  variable  indices,  as  La- 
place has  done  in  his  "  Analytical  Theory  of  Probabili- 
ties," by  the  analysis  to  which  he  has  given  the  name 
of  Theory  of  generating1  Functions,  although  it  is  real- 
ly only  a  new  and  higher  branch  of  the  calculus  of  finite 
differences  or  of  the  general  theory  of  series. 

These  general  views  which  I  have  indicated  give  only 
an  imperfect  idea  of  the  truly  infinite  extent  and  variety 
of  the  questions  to  which  geometers  have  risen  by  means 
of  this  single  consideration  of  series,  so  simple  in  ap- 
pearance and  so  limited  in  its  origin.  It  necessarily 
presents  as  many  different  cases  as  the  algebraic  resolu- 
tion of  equations,  considered  in  its  whole  extent ;  and  it 
is,  by  its  nature,  much  more  complicated,  so  much,  in- 
deed, that  it  always  needs  this  last  to  conduct  it  to  a  com- 
plete solution.  We  may,  therefore,  anticipate  what  must 
still  be  its  extreme  imperfection,  in  spite  of  the  successive 
labours  of  several  geometers  of  the  first  order.  We  do 
not,  indeed,  possess  as  yet  the  complete  and  logical  solu- 
tion of  any  but  the  most  simple  questions  of  this  na- 
ture. 


172  THE   CALCULUS   OF   FINITE   DIFFERENCES. 

Its  identity  with  this  Calculus.  It  is  now  easy  to 
conceive  the  necessary  and  perfect  identity,  which  has 
been  already  announced,  between  the  calculus  of  finite 
differences  and  the  theory  of  series  considered  in  all  its 
bearings.  In  fact,  every  differentiation  after  the  man- 
ner of  Taylor  evidently  amounts  to  finding  the  law  of 
formation  of  a  series  with  one  or  with  several  variable 
indices,  from  the  expression  of  its  general  term ;  in  the 
same  way,  every  analogous  integration  may  be  regard- 
ed as  having  for  its  object  the  summation  of  a  series,  the 
general  term  of  which  would  be  expressed  by  the  pro- 
posed difference.  In  this  point  of  view,  the  various  prob- 
lems of  the-  calculus  of  differences,  direct  or  inverse,  re- 
solved by  Taylor  and  his  successors,  have  really  a  very 
great  value,  as  treating  of  important  questions  relating 
to  series.  But  it  is  very  doubtful  if  the  form  and  the 
notation  introduced  by  Taylor  really  give  any  essential 
facility  in  the  solution  of  questions  of  this  kind.  It 
would  be,  perhaps,  more  advantageous  for  most  cases,  and 
certainly  more  logical,  to  replace  the  differences  by  the 
terms  themselves,  certain  combinations  of  which  they 
represent.  As  the  calculus  of  Taylor  does  not  rest  on 
a  truly  distinct  fundamental  idea,  and  has  nothing  pecu- 
liar to  it  but  its  system  of  signs,  there  could  never  really 
be  any  important  advantage  in  considering  it  as  detached 
from  ordinary  analysis,  of  which  it  is,  in  reality,  only  an 
immense  branch.  This  consideration  of  differences,  most 
generally  useless,  even  if  it  does  not  cause  complication, 
seems  to  me  to  retain  the  character  of  an  epoch  in  which, 
analytical  ideas  not  being  sufficiently  familiar  to  geome- 
ters, they  were  naturally  led  to  prefer  the  special  forms 
suitable  for  simple  numerical  comparisons. 


APPLICATIONS.  173 

PERIODIC  OR  DISCONTINUOUS  FUNCTIONS.       • 

However  that  may  be,  I  must  not  finish  this  general 
appreciation  of  the  calculus  of  finite  differences  without 
noticing  a  new  conception  to  which  it  has  given  birth,  and 
which  has  since  acquired  a  great  importance.  It  is  the 
consideration  of  those  periodic  or  discontinuous  functions 
which  preserve  the  same  value  for  an  infinite  series  of 
values  of  the  corresponding  variables,  subjected  to  a  cer- 
tain law,  and  which  must  be  necessarily  added  to  the  in- 
tegrals of  the  equations  of  finite  differences  in  order  to 
render  them  sufficiently  general,  as  simple  arbitrary  con- 
stants are  added  to  all  quadratures  in  order  to  complete 
their  generality.  This  idea,  primitively  introduced  by 
Euler,  has  since  been  the  subject  of  extended  investiga- 
tion by  M.  Fourier,  who  has  made  new  and  important 
applications  of  it  in  his  mathematical  theory  of  heat. 

APPLICATIONS  OF  THIS  CALCULUS. 

Series.  Among  the  principal  general  applications 
which  have  been  made  of  the  calculus  of  finite  differen- 
ces, it  would  be  proper  to  place  in  the  first  rank,  as  the 
most  extended  and  the  most  important,  the  solution  of 
questions  relating  to  series ;  if,  as  has  been  shown,  the 
general  theory  of  series  ought  not  to  be  considered  as  con- 
stituting, by  its  nature,  the  actual  foundation  of  the  cal- 
culus of  Taylor. 

Interpolations.  This  great  class  of  problems  being 
then  set  aside,  the  most  essential  of  the  veritable  appli- 
cations of  the  analysis  of  Taylor  is,  undoubtedly,  thus 
far,  the  general  method  of  interpolations,  so  frequently 
and  so  usefully  employed  in  the  investigation  of  the  em- 


174  THE   CALCULUS   OF  FINITE   DIFFEllLNCES. 

pirical  laws  of  natural  phenomena.  The  question  consists, 
as  is  well  known,  in  intercalating  between  certain  given 
numbers  other  intermediate  numbers,  subjected  to  the 
same  law  which  we  suppose  to  exist  between  the  first. 
We  can  abundantly  verify,  in  this  principal  application 
of  the  calculus  of  Taylor,  how  truly  foreign  and  often  in- 
convenient is  the  consideration  of  differences  with  respect 
to  the  questions  which  depend  on  that  analysis.  Indeed, 
Lagrange  has  replaced  the  formulas  of  interpolation,  de- 
duced from  the  ordinary  algorithm  of  the  calculus  of 
finite  differences,  by  much  simpler  general  formulas, 
which  are  now  almost  always  preferred,  and  which  have 
been  found  directly,  without  making  any  use  of  the  no- 
tion of  differences,  which  only  complicates  the  question. 
Approximate  Rectification,  Sfc.  A  last  important 
class  of  applications  of  the  calculus  of  finite  differences, 
which  deserves  to  be  distinguished  from  the  preceding, 
consists  in  the  eminently  useful  employment  made  of  it 
in  geometry  for  determining  by  approximation  the  length 
and  the  area  of  any  curve,  and  in  the  same  way  the  cu- 
bature  of  a  body  of  any  form  whatever.  This  procedure 
(which  may  besides  be  conceived  abstractly  as  depending 
on  the  same  analytical  investigation  as  the  question  of 
interpolation)  frequently  offers  a  valuable  supplement  to 
the  entirely  logical  geometrical  methods  which  often  lead 
to  integrations,  which  we  do  not  yet  know  how  to  effect, 
or  to  calculations  of  very  complicated  execution. 

Such  are  the  various  principal  considerations  to  be 
noticed  with  respect  to  the  calculus  of  finite  differences. 
This  examination  completes  the  proposed  philosophical 
outline  of  ABSTRACT  MATHEMATICS. 


CONCRETE    MATHEMATICS.  175 

CONCRETE  MATHEMATICS  will  now  be  the  subject  of  a 
similar  labour.  In  it  we  shall  particularly  devote  our- 
selves to  examining  how  it  has  been  possible  (supposing 
the  general  science  of  the  calculus  to  be  perfect),  by  inva- 
riable procedures,  to  reduce  to  pure  questions  of  analysis 
all  the  problems  which  can  be  presented  by  Geometry  and 
Mechanics,  and  thus  to  impress  on  these  two  fundamental 
bases  of  natural  philosophy  a  degree  of  precision  and  es- 
pecially of  unity  ;  in  a  word,  a  character  of  high  perfec- 
tion, which  could  be  communicated  to  them  by  such  a 
course  alone. 


BOOK   II. 


GEOMETRY. 


BOOK    II. 
GEOMETRY. 


CHAPTER    I. 

GENERAL  VIEW  OF  GEOMETRY. 

Its  true  Nature.  After  the  general  exposition  of  the 
philosophical  character  of  concrete  mathematics,  com- 
pared with  that  of  abstract  mathematics,  given  in  the  in* 
troductory  chapter,  it  need  not  here  be  shown  in  a  special 
manner  that  geometry  must  be  considered  as  a  true  nat- 
ural science,  only  much  more  simple,  and  therefore  much 
more  perfect,  than  any  other.  This  necessary  perfection 
of  geometry,  obtained  essentially  by  the  application  of 
mathematical  analysis,  which  it  so  eminently  admits,  is 
apt  to  produce  erroneous  views  of  the  real  nature  of  this 
fundamental  science,  which  most  minds  at  present  con- 
ceive to  be  a  purely  logical  science  quite  independent  of 
observation.  It  is  nevertheless  evident,  to  any  one  who 
examines  with  attention  the  character  of  geometrical  rea- 
sonings, even  in  the  present  state  of  abstract  geometry, 
that,  although  the  facts  which  are  considered  in  it  are 
much  more  closely  united  than  those  relating  to  any  other 
science,  still  there  always  exists,  with  respect  to  every 
body  studied  by  geometers,  a  certain  number  of  primitive 
phenomena,  which,  since  they  are  not  established  by  any 


jg0  GEOMETRY. 

reasoning,  must  be  founded  on  observation  alone,  and 
which  form  the  necessary  basis  of  all  the  deductions. 

The  scientific  superiority  of  geometry  arises  from  the 
phenomena  which  it  considers  being  necessarily  the  most 
universal  and  the  most  simple  of  all.  Not  only  may  all 
the  bodies  of  nature  give  rise  to  geometrical  inquiries,  as 
well  as  mechanical  ones,  but  still  farther,  geometrical 
phenomena  would  still  exist,  even  though  all  the  parts 
of  the  universe  should  be  considered  as  immovable.  Ge- 
ometry is  then,  by  its  nature,  more  general  than  mechan- 
ics. At  the  same  time,  its  phenomena  are  more  simple, 
for  they  are  evidently  independent  of  mechanical  phenom- 
ena, while  these  latter  are  always  complicated  with  the 
former.  The  same  relations  hold  good  in  comparing 
geometry  with  abstract  thermology. 

For  these  reasons,  in  our  classification  we  have  made 
geometry  the  first  part  of  concrete  mathematics ;  that 
part  the  study  of  which,  in  addition  to  its  own  impor- 
tance, serves  as  the  indispensable  basis  of  all  the  rest. 

Before  considering  directly  the  philosophical  study  of 
the  different  orders  of  inquiries  which  constitute  our 
present  geometry,  we  should  obtain  a  clear  and  exact 
idea  of  the  general  destination  of  that  science,  viewed  in 
all  its  bearings.  Such  is  the  object  of  this  chapter. 

Definition.  Geometry  is  commonly  defined  in  a  very 
vague  and  entirely  improper  manner,  as  being  the  science 
of  extension.  An  improvement  on  this  would  be  to  say 
that  geometry  has  for  its  object  the  measurement  of  ex- 
tension ;  but  such  an  explanation  would  be  very  insuf- 
ficient, although  at  bottom  correct,  and  would  be  far  from 
giving  any  idea  of  the  true  general  character  of  geomet- 
rical science 


THE  IDEA   OF  SPACE.  1QJ 

To  do  this,  I  think  that  I  should  first  explain  two  fun- 
damental ideas,  which,  very  simple  in  themselves,  have 
been  singularly  obscured  by  the  employment  of  meta- 
physical considerations. 

The  Idea  of  Space.  The  first  is  that  of  Space. 
This  conception  properly  consists  simply  in  this,  that,  in- 
stead of  considering  extension  in  the  bodies  themselves, 
we  view  it  in  an  indefinite  medium,  which  we  regard  as 
containing  all  the  bodies  of  the  universe.  This  notion  is 
naturally  suggested  to  us  by  observation,  when  we  think 
of  the  impression  which  a  body  would  leave  in  a  fluid  in 
which  it  had  been  placed.  It  is  clear,  in  fact,  that,  as  re- 
gards its  geometrical  relations,  such  an  impression  may 
be  substituted  for  the  body  itself,  without  altering  the 
reasonings  respecting  it.  As  to  the  physical  nature  of 
this  indefinite  space,  we  are  spontaneously  led  to  repre- 
sent it  to  ourselves,  as  being  entirely  analogous  to  the 
actual  medium  in  which  we  live ;  so  that  if  this  me- 
dium was  liquid  instead  of  gaseous,  our  geometrical  space 
would  undoubtedly  be  conceived  as  liquid  also.  This 
circumstance  is,  moreover,  only  very  secondary,  the  es- 
sential object  of  such  a  conception  being  only  to  make 
us  view  extension  separately  from  the  bodies  which  man- 
ifest it  to  us.  We  can  easily  understand  in  advance  the 
importance  of  this  fundamental  image,  since  it  permits- 
us  to  study  geometrical  phenomena  in  themselves,  ab- 
straction being  made  of  all  the  other  phenomena  which 
constantly  accompany  them  in  real  bodies,  without,  how- 
ever, exerting  any  influence  over  them.  The  regular  es- 
tablishment of  this  general  abstraction  must  be  regard- 
ed as  the  first  step  which  has  been  made  in  the  rational 
study  of  geometry,  which  would  have  been  impossible  if 


182 


GEOMETRY. 


it  had  been  necessary  to  consider,  together  with  the  form 
and  the  magnitude  of  bodies,  all  their  other  physical 
properties.  The  use  of  such  an  hypothesis,  which  is 
perhaps  the  most  ancient  philosophical  conception  crea- 
ted by  the  human  mind,  has  now  become  so  familiar  to 
us,  that  we  have  difficulty  in  exactly  estimating  its  im- 
portance, by  trying  to  appreciate  the  consequences  which 
would  result  from  its  suppression. 

Different  Kinds  of  Extension.  The  second  prelimi- 
nary geometrical  conception  which  we  have  to  examine 
is  that  of  the  different  kinds  of  extension,  designated  by 
the  words  volume,  surface,  line,  and  even  point,  and  of 
which  the  ordinary  explanation  is  so  unsatisfactory.^ 

Although  it  is  evidently  impossible  to  conceive  any  ex- 
tension absolutely  deprived  of  any  one  of  the  three  fun- 
damental dimensions,  it  is  no  less  incontestable  that,  in 
a  great  number  of  occasions,  even  of  immediate  utility, 
geometrical  questions  depend  on  only  two  dimensions, 
considered  separately  from  the  third,  or  on  a  single  dimen- 
sion, considered  separately  from  the  two  others.  Again, 
independently  of  this  direct  motive,  the  study  of  exten- 
sion with  a  single  dimension,  and  afterwards  with  two, 
clearly  presents  itself  as  an  indispensable  preliminary  for 
facilitating  the  study  of  complete  bodies  of  three  dimen- 
sions, the  immediate  theory  of  which  would  be  too  com- 

*  Lacroix  has  justly  criticised  the  expression  of  solid,  commonly  used  by 
geometers  to  designate  a  volume.  It  is  certain,  in  fact,  that  when  we  wish 
to  consider  separately  a  certain  portion  of  indefinite  space,  conceived  as  gas- 
eous, we  mentally  solidify  its  exterior  envelope,  so  that  a  line  and  a  surface 
are  habitually,  to  our  minds,  just  as  solid  as  a  volume.  It  may  also  be  re- 
marked that  most  generally,  in  order  that  bodies  may  penetrate  one  another 
with  more  facility,  we  are  obliged  to  imagine  the  interior  of  the  volumes  to 
be  hollow,  which  renders  still  more  sensible  the  impropriety  of  the  word 
to  Lid. 


DIFFERENT  KINDS    OF  EXTENSION.      193 

plicated.  Such  are  the  two  general  motives  which  oblige 
geometers  to  consider  separately  extension  with  regard  to 
one  or  to  two  dimensions,  as  well  as  relatively  to  all  three 
together. 

The  general  notions  of  surface  and  of  line  have  been 
formed  by  the  human  mind,  in  order  that  it  may  be  able 
to  think,  in  a  permanent  manner,  of  extension  in  two 
directions,  or  in  one  only.  The  hyperbolical  expressions 
habitually  employed  by  geometers  to  define  these  notions 
tend  to  convey  false  ideas  of  them ;  but,  examined  in 
themselves,  they  have  no  other  object  than  to  permit  us 
to  reason  with  facility  respecting  these  two  kinds  of  ex- 
tension, making  complete  abstraction  of  that  which  ought 
not  to  be  taken  into  consideration.  Now  for  this  it  is 
sufficient  to  conceive  the  dimension  which  we  wish  to 
eliminate  as  becoming  gradually  smaller  and  smaller, 
the  two  others  remaining  the  same,  until  it  arrives  at 
such  a  degree  of  tenuity  that  it  can  no  longer  fix  the  at- 
tention. It  is  thus  that  we  naturally  acquire  the  rea.1 
idea  of  a  surface,  and,  by  a  second  analogous  operation, 
the  idea  of  a  line,  by  repeating  for  breadth  what  we  had 
at  first  done  for  thickness.  Finally,  if  we  again  repeat 
the  same  operation,  we  arrive  at  the  idea  of  a  point,  or 
of  an  extension  considered  only  with  reference  to  its 
place,  abstraction  being  made  of  all  magnitude,  and  de- 
signed consequently  to  determine  positions. 

Surfaces  evidently  have,  moreover,  the  general  prop- 
erty of  exactly  circumscribing  volumes ;  and  in  the  same 
way,  lines,  in  their  turn,  circumscribe  surfaces  and  are 
limited  by  points.  But  this  consideration,  to  which  too 
much  importance  is  often  given,  is  only  a  secondary 
one. 


184 


GEOMETRY. 


Surfaces  and  lines  are,  then,  in  reality,  always  con- 
ceived  with  three  dimensions ;  it  would  be,  in  fact,  im- 
possible to  represent  to  one's  self  a  surface  otherwise  than 
as  an  extremely  thin  plate,  and  a  line  otherwise  than  as 
an  infinitely  fine  thread.  It  is  even  plain  that  the  de- 
gree of  tenuity  attributed  by  each  individual  to  the  di- 
mensions of  which  he  wishes  to  make  abstraction  is  not 
constantly  identical,  for  it  must  depend  on  the  degree  of 
subtilty  of  his  habitual  geometrical  observations.  This 
want  of  uniformity  has,  besides,  no  real  inconvenience, 
since  it  is  sufficient,  in  order  that  the  ideas  of  surface 
and  of  line  should  satisfy  the  essential  condition  of  their 
destination,  for  each  one  to  represent  to  himself  the  di- 
mensions which  are  to  be  neglected  as  being  smaller  than 
all  those  whose  magnitude  his  daily  experience  gives  him 
occasion  to  appreciate. 

We  hence  see  how  devoid  of  all  meaning  are  the  fan- 
tastic discussions  of  metaphysicians  upon  the  foundations 
of  geometry.  It  should  also  be  remarked  that  these  pri- 
mordial ideas  are  habitually  presented  by  geometers  in 
an  unphilosophical  manner,  since,  for  example,  they  ex- 
plain the  notions  of  the  different  sorts  of  extent  in  an 
order  absolutely  the  inverse  of  their  natural  dependence, 
which  often  produces  the  most  serious  inconveniences  in 
elementary  instruction. 

THE  FINAL  OBJECT  OF  GEOMETRY. 

These  preliminaries  being  established,  we  can  proceed 
directly  to  the  general  definition  of  geometry,  continuing 
to  conceive  this  science  as  having  for  its  final  object  the 
measurement  of  extension. 

It  is  necessary  in  this  matter  to  go  into  a  thorough 


MEASUREMENT  OF  SURFACES,  ETC.   185 

explanation,  founded  on  the  distinction  of  the  three  kinds 
of  extension,  since  the  notion  of  measurement  is  not  ex- 
actly the  same  with  reference  to  surfaces  and  volumes 
as  to  lines. 

Nature  of  Geometrical  Measurement.  If  we  take  the 
word  measurement  in  its  direct  and  general  mathemat- 
ical acceptation,  which  signifies  simply  the  determina- 
tion of  the  value  of  the  ratios  between  any  homogeneous 
magnitudes,  we  must  consider,  in  geometry,  that  the 
measurement  of  surfaces  and  of  volumes,  unlike  that  of 
lines,  is  never  conceived,  even  in  the  most  simple  and  the 
most  favourable  cases,  as  being  effected  directly.  The 
comparison  of  two  lines  is  regarded  as  direct ;  that  of 
two  surfaces  or  of  two  volumes  is,  on  the  contrary,  al- 
ways indirect.  Thus  we  conceive  that  two  lines  may 
be  superposed;  but  the  superposition  of  two  surfaces,  or, 
still  more  so,  of  two  volumes,  is  evidently  impossible  in 
most  cases  ;  and,  even  when  it  becomes  rigorously  prac- 
ticable, such  a  comparison  is  never  either  convenient  or 
exact.  It  is,  then,  very  necessary  to  explain  wherein 
properly  consists  the  truly  geometrical  measurement  of 
a  surface  or  of  a  volume. 

Measurement  of  Surfaces  and  of  Volumes.  For  this 
we  must  consider  that,  whatever  may  be  the  form  of  a 
body,  there  always  exists  a  certain  number  of  lines,  more 
or  less  easy  to  be  assigned,  the  length  of  which  is  suffi- 
cient to  define  exactly  the  magnitude  of  its  surface  or  of 
its  volume.  Geometry,  regarding  these  lines  as  alone 
susceptible  of  being  directly  measured,  proposes  to  deduce, 
from  the  simple  determination  of  them,  the  ratio  of  the 
surface  or  of  the  volume  sought,  to  the  unity  of  surface, 
or  to  the  unity  of  volume.  Thus  the  general  object  of 


186 


GEOMETRY. 


geometry,  with  respect  to  surfaces  and  to  volumes,  is 
properly  to  reduce  all  comparisons  of  surfaces  or  of  vol- 
umes to  simple  comparisons  of  lines. 

Besides  the  very  great  facility  which  such  a  transform- 
ation evidently  offers  for  the  measurement  of  volumes 
and  of  surfaces,  there  results  from  it,  in  considering  it 
in  a  more  extended  and  more  scientific  manner,  the  gen- 
eral possibility  of  reducing  to  questions  of  lines  all  ques- 
tions relating  to  volumes  and  to  surfaces,  considered  with 
reference  to  their  magnitude.  Such  is  often  the  most 
important  use  of  the  geometrical  expressions  which  de- 
termine surfaces  and  volumes  in  functions  of  the  corre- 
sponding lines. 

It  is  true  that  direct  comparisons  between  surfaces  or 
between  volumes  are  sometimes  employed  ;  but  such 
measurements  are  not  regarded  as  geometrical,  but  only 
as  a  supplement  sometimes  necessary,  although  too  rare- 
ly applicable,  to  the  insufficiency  or  to  the  difficulty  of 
truly  rational  methods.  It  is  thus  that  we  often  deter- 
mine the  volume  of  a  body,  and  in  certain  cases  its  sur- 
face, by  means  of  its  weight.  In  the  same  way,  on  othei 
occasions,  when  we  can  substitute  for  the  proposed  vol. 
ume  an  equivalent  liquid  volume,  we  establish  directly 
the  comparison  of  the  two  volumes,  by  profiting  by  the 
property  possessed  by  liquid  masses,  of  assuming  any  de- 
sired form.  But  all  means  of  this  nature  are  purely  me- 
chanical, and  rational  geometry  necessarily  rejects  them. 

To  render  more  sensible  the  difference  between  these 
modes  of  determination  and  true  geometrical  measure- 
ments, I  will  cite  a  single  very  remarkable  example  ;  the 
manner  in  which  Galileo  determined  the  ratio  of  the  or- 
dinary cycloid  to  that  of  the  generating  circle.  The 


MEASUREMENT   OF  LINES.  1Q7 

geometry  of  his  time  was  as  yet  insufficient  for  the  ra- 
tional solution  of  such  a  problem.  Galileo  conceived 
the  idea  of  discovering  that  ratio  by  a  direct  experiment. 
Having  weighed  as  exactly  as  possible  two  plates  of  the 
same  material  and  of  equal  thickness,  one  of  them  hav- 
ing the  form  of  a  circle  and  the  other  that  of  the  gener- 
ated cycloid,  he  found  the  weight  of  the  latter  always 
triple  that  of  the  former ;  whence  he  inferred  that  the 
area  of  the  cycloid  is  triple  that  of  the  generating  circle, 
a  result  agreeing  with  the  veritable  solution  subsequent- 
ly obtained  by  Pascal  and  Wallis.  Such  a  success  ev- 
idently depends  on  the  extreme  simplicity  of  the  ratio 
sought;  and  we  can  understand  the  necessary  insufficien- 
cy of  such  expedients,  even  when  they  are  actually  prac- 
ticable. 

We  see  clearly,  from  what  precedes,  the  nature  of  that 
part  of  geometry  relating  to  volumes  and  that  relating  to 
surfaces.  But  the  character  of  the  geometry  of  lines  is 
not  so  apparent,  since,  in  order  to  simplify  the  exposition, 
we  have  considered  the  measurement  of  lines  as  being 
made  directly.  There  is,  therefore,  needed  a  comple- 
mentary explanation  with  respect  to  them. 

Measurement  of  curved  Lines.  For  this  purpose,  it 
js  sufficient  to  distinguish  between  the  right  -line  and 
curved  lines,  the  measurement  of  the  first  being  alone 
regarded  as  direct,  and  that  of  the  other  as  always  indi- 
rect. Although  superposition  is  sometimes  strictly  prac- 
ticable for  curved  lines,  it  is  nevertheless  evident  that 
truly  rational  geometry  must  necessarily  reject  it,  as 
not  admitting  of  any  precision,  even  when  it  is  possible. 
The  geometry  of  lines  has,  then,  for  its  general  object,  to 
reduce  in  every  case  the  measurement  of  curved  lines  to 


GEOMETRY. 

that  of  right  lines ;  and  consequently,  in  the  most  ex- 
tended point  of  view,  to  reduce  to  simple  questions  of 
right  lines  all  questions  relating  to  the  magnitude  of  any 
curves  whatever.  To  understand  the  possibility  of  such 
a  transformation,  we  must  remark,  that  in  every  curve 
there  always  exist  certain  right  lines,  the  length  of  which 
must  be  sufficient  to  determine  that  of  the  curve.  Thus, 
in  a  circle,  it  is  evident  that  from  the  length  of  the  ra- 
dius we  must  be  able  to  deduce  that  of  the  circumfer- 
ence ;  in  the  same  way,  the  length  of  an  ellipse  depends 
on  that  of  its  two  axes  ;  the  length  of  a  cycloid  upon  the 
diameter  of  the  generating  circle,  &c.  ;  and  if,  instead 
of  considering  the  whole  of  each  curve,  we  demand,  more 
generally,  the  length  of  any  arc,  it  will  be  sufficient  to 
add  to  the  different  rectilinear  parameters,  which  deter- 
mine the  whole  curve,  the  chord  of  the  proposed  arc,  or 
the  co-ordinates  of  its  extremities.  To  discover  the  re- 
lation which  exists  between  the  length  of  a  curved  line 
and  that  of  similar  right  lines,  is  the  general  problem  of 
the  part  of  geometry  which  relates  to  the  study  of  lines. 
Combining  this  consideration  with  those  previously 
suggested  with  respect  to  volumes  and  to  surfaces,  we 
may  form  a  very  clear  idea  of  the  science  of  geometry, 
conceived  in  all  its  parts,  by  assigning  to  it,  for  its  gen- 
eral object,  the  final  reduction  of  the  comparisons  of  all 
kinds  of  extent,  volumes,  surfaces,  or  lines,  to  simple  com- 
parisons of  right  lines,  the  only  comparisons  regarded  as 
capable  of  being  made  directly,  and  which  indeed  could 
not  be  reduced  to  any  others  more  easy  to  effect.  Such 
a  conception,  at  the  same  time,  indicates  clearly  the  ver- 
itable character  of  geometry,  and  seems  suited  to  show 
at  a  single  glance  its  utility  and  its  perfection. 


MEASUREMENT   OF   LINES.  1Q9 

Measurement  of  right  Lines.  In  order  to  complete 
this  fundamental  explanation,  I  have  yet  to  show  how 
there  can  be,  in  geometry,  a  special  section  relating  to 
the  right  line,  which  seems  at  first  incompatible  with  the 
principle  that  the  measurement  of  this  class  of  lines  must 
always  be  regarded  as  direct. 

It  is  so,  in  fact,  as  compared  with  that  of  curved  lines, 
and  of  all  the  other  objects  which  geometry  considers. 
But  it  is  evident  that  the  estimation  of  a  right  line  can- 
not be  viewed  as  direct  except  so  far  as  the  linear  unit  can 
be  applied  to  it.  Now  this  often  presents  insurmount- 
able difficulties,  as  I  had  occasion  to  show,  for  another 
reason,  in  the  introductory  chapter.  We  must,  then, 
make  the  measurement  of  the  proposed  right  line  depend 
on  other  analogous  measurements  capable  of  being  effect- 
ed directly.  There  is,  then,  necessarily  a  primary  dis- 
tinct branch  of  geometry,  exclusively  devoted  to  the  right 
line ;  its  object  is  to  determine  certain  right  lines  from 
others  by  means  of  the  relations  belonging  to  the  figures 
resulting  from  their  assemblage.  This  preliminary  part 
of  geometry,  which  is  almost  imperceptible  in  viewing 
the  whole  of  the  science,  is  nevertheless  susceptible  of  a 
great  development.  It  is  evidently  of  especial  import- 
ance, since  all  other  geometrical  measurements  are  refer- 
red to  those  of  right  lines,  and  if  they  could  not  be  de- 
termined, the  solution  of  every  question  would  remain 
unfinished. 

Such,  then,  are  the  various  fundamental  parts  of  ra- 
tional geometry,  arranged  according  to  their  natural  de- 
pendence ;  the  geometry  of  lines  being  first  considered, 
beginning  with  the  right  line  ;  then  the  geometry  of  sur- 
faces, and,  finally,  that  of  solids. 


190 


GEOMETRY. 


INFINITE  EXTENT  OF   ITS  FIELD. 

Having  determined  with  precision  the  general  and 
final  object  of  geometrical  inquiries,  the  science  musl 
now  be  considered  with  respect  to  the  field  embraced  by 
each  of  its  three  fundamental  sections. 

Thus  considered,  geometry  is  evidently  susceptible, 
by  its  nature,  of  an  extension  which  is  rigorously  in- 
finite ;  for  the  measurement  of  lines,  of  surfaces,  or 
of  volumes  presents  necessarily  as  many  distinct  ques- 
tions as  we  can  conceive  different  figures  subjected  to 
exact  definitions ;  and  their  number  is  evidently  infi- 
nite. 

Geometers  limited  themselves  at  first  to  consider  the 
most  simple  figures  which  were  directly  furnished  them 
by  nature,  or  which  were  deduced  from  these  primitive 
elements  by  the  least  complicated  combinations.  But 
they  have  perceived,  since  Descartes,  that,  in  order  to  con- 
stitute the  science  in  the  most  philosophical  manner,  it 
was  necessary  to  make  it  apply  to  all  imaginable  figures. 
This  abstract  geometry  will  then  inevitably  comprehend 
as  particular  cases  all  the  different  real  figures  which 
the  exterior  world  could  present.  It  is  then  a  fundamen- 
tal principle  in  truly  rational  geometry  to  consider,  as 
far  as  possible,  all  figures  which  can  be  rigorously  conr 
ceived. 

The  most  superficial  examination  is  enough  to  con- 
vince us  that  these  figures  present  a  variety  which  is 
quite  infinite. 

Infinity  of  Lines.  With  respect  to  curved  lines,  re- 
garding them  as  generated  by  the  motion  of  a  point  gov- 
erned by  a  certain  law,  it  is  plain  that  we  shall  have,  in 


ITS  INFINITE  EXTENT. 

general,  as  many  different  curves  as  we  conceive  differ- 
ent laws  for  this  motion,  which  may  evidently  be  deter- 
mined by  an  infinity  of-distinct  conditions ;  although  it 
may  sometimes  accidentally  happen  that  new  generations 
produce  curves  which  have  been  already  obtained.  Thus, 
among  plane  curves,  if  a  point  moves  so  as  to  remain  con- 
stantly at  the  same  distance  from  a  fixed  point,  it  will 
generate  a  circle  ;  if  it  is  the  sum  or  the  difference  of 
its  distances  from  two  fixed  points  which  remains  con- 
stant, the  curve  described  will  be  an  ellipse  or  an  hyper- 
bola ;  if  it  is  their  product,  we  shall  have  an  entirely  dif- 
ferent curve ;  if  the  point  departs  equally  from  a  fixed 
point  and  from  a  fixed  line,  it  will  describe  a  parabola; 
if  it  revolves  on  a  circle  at  the  same  time  that  this  cir- 
cle rolls  along  a  straight  line,  we  shall  have  a  cycloid; 
if  it  advances  along  a  straight  line,  while  this  line,  fixed 
at  one  of  its  extremities,  turns  in  any  manner  whatever, 
there  will  result  what  in  general  terms  are  called  spi- 
rals, which  of  themselves  evidently  present  as  many 
perfectly  distinct  curves  as  we  can  suppose  different  re- 
lations between  these  two  motions  of  translation  and  of 
rotation,  &c.  Each  of  these  different  curves  may  then 
furnish  new  ones,  by  the  different  general  constructions 
which  geometers  have  imagined,  and  which  give  rise  to 
evolutes,  to  epicycloids,  to  caustics,  &c.  Finally,  there 
exists  a  still  greater  variety  among  curves  of  double  cur- 
vature. 

Infinity  of  Surfaces.  As  to  surfaces,  the  figures  are 
necessarily  more  different  still,  considering  them  as  gen- 
erated by  the  motion  of  lines.  Indeed,  the  figure  may 
then  vary,  not  only  in  considering,  as  in  curves,  the  dif- 
ferent infinitely  numerous  laws  to  which  the  motion  of 


GEOMETRY. 

the  generating  line  may  be  subjected,  but  also  in  sup- 
posing that  this  line  itself  may  change  its  nature ;  a  cir- 
cumstance which  has  nothing  analogous  in  curves,  since 
the  points  which  describe  them  cannot  have  any  distinct 
figure.  Two  classes  of  very  different  conditions  may 
then  cause  the  figures  of  surfaces  to  vary,  while  there 
exists  only  one  for  lines.  It  is  useless  to  cite  examples 
of  this  doubly  infinite  multiplicity  of  surfaces.  It  would 
be  sufficient  to  consider  the  extreme  variety  of  the  single 
group  of  surfaces  which  may  be  generated  by  a  right  line, 
and  which  comprehends  the  whole  family  of  cylindrical 
surfaces,  that  of  conical  surfaces,  the  most  general  class 
of  developable  surfaces,  &c. 

Infinity  of  Volumes.  With  respect  to  volumes,  there 
is  no  occasion  for  any  special  consideration,  since  they  are 
distinguished  from  each  other  only  by  the  surfaces  which 
bound  them. 

In  order  to  complete  this  sketch,  it  should  be  added 
that  surfaces  themselves  furnish  a  new  general  means  of 
conceiving  new  curves,  since  every  curve  may  be  regard- 
ed as  produced  by  the  intersection  of  two  surfaces.  It 
is  in  this  way,  indeed,  that  the  first  lines  which  we  may 
regard  as  having  been  truly  invented  by  geometers  were 
obtained,  since  nature  gave  directly  the  straight  line  and 
the  circle.  We  know  that  the  ellipse,  the  parabola,  and 
the  hyperbola,  the  only  curves  completely  studied  by  the 
ancients,  were  in  their  origin  conceived  only  as  result- 
ing from  the  intersection  of  a  cone  with  circular  base  by 
a  plane  in  different  positions.  It  is  evident  that,  by  the 
combined  employment  of  these  different  general  means 
for  the  formation  of  lines  and  of  surfaces,  we  could  pro- 
duce a  rigorously  infinitely  series  of  distinct  forms  in 


EXPANSION   OF   ORIGINAL   DEFINITION.     193 

starting  from  only  a  very  small  number  of  figures  di- 
rectly furnished  by  observation. 

Analytical  invention  of  Curves,  Sfc.  Finally,  all 
the  various  direct  means  for  the  invention  of  figures 
have  scarcely  any  farther  importance,  since  rational  ge- 
ometry has  assumed  its  final  character  in  the  hands  of 
Descartes.  Indeed,  as  we  shall  see  more  fully  in  chap- 
ter iii.,  the  invention  of  figures  is  now  reduced  to  the 
invention  of  equations,  so  that  nothing  is  more  easy  than 
to  conceive  new  lines  and  new  surfaces,  by  changing  at 
will  the  functions  introduced  into  the  equations.  This 
simple  abstract  procedure  is,  in  this  respect,  infinitely 
more  fruitful  than  all  the  direct  resources  of  geometry,  de- 
veloped by  the  most  powerful  imagination,  which  should 
devote  itself  exclusively  to  that  order  of  conceptions.  It 
also  explains,  in  the  most  general  and  the  most  striking 
manner,  the  necessarily  infinite  variety  of  geometrical 
forms,  which  thus  corresponds  to  the  diversity  of  analyt- 
ical functions.  Lastly,  it  shows  no  less  clearly  that  the 
different  forms  of  surfaces  must  be  still  more  numerous 
than  those  of  lines,  since  lines  are  represented  analyti- 
cally by  equations  with  two  variables,  while  surfaces  give 
rise  to  equations  with  three  variables,  which  necessarily 
present  a  greater  diversity. 

The  preceding  considerations  are  sufficient  to  show 
clearly  the  rigorously  infinite  extent  of  each  of  the  three 
general  sections  of  geometry. 

EXPANSION   OF    ORIGINAL   DEFINITION. 

To  complete  the  formation  of  an  exact  and  sufficient- 
ly extended  idea  of  the  nature  of  geometrical  inquiries, 
it  is  now  indispensable  to  return  to  the  general  definition 

N 


194  GEOMETRY. 

above  given,  in  order  to  present  it  under  a  new  point  of 
view,  without  which  the  complete  science  would  be  only 
very  imperfectly  conceived. 

When  we  assign  as  the  object  of  geometry  the  meas- 
urement of  all  sorts  of  lines,  surfaces,  and  volumes,  that 
is,  as  has  been  explained,  the  reduction  of  all  geometri- 
cal comparisons  to  simple  comparisons  of  right  lines,  we 
have  evidently  the  advantage  of  indicating  a  general  des- 
tination very  precise  and  very  easy  to  comprehend.  But 
if  we  set  aside  every  definition,  and  examine  the  actual 
composition  of  the  science  of  geometry,  we  will  at  first 
be  induced  to  regard  the  preceding  definition  as  much 
too  narrow  ;  for  it  is  certain  that  the  greater  part  of  the 
investigations  which  constitute  our  present  geometry  do 
not  at  all  appear  to  have  for  their  object  the  measure- 
ment of  extension.  In  spite  of  this  fundamental  objec- 
tion, I  will  persist  in  retaining  this  definition  ;  for,  in 
fact,  if,  instead  of  confining  ourselves  to  considering  the 
different  questions  of  geometry  isofatedly,  we  endeavour 
to  grasp  the  leading  questions,  in  comparison  with  which 
all  others,  however  important  they  may  be,  must  be  re- 
garded as  only  secondary,  we  will  finally  recognize  that 
the  measurement  of  lines,  of  surfaces,  and  of  volumes,  is 
the  invariable  object,  sometimes  direct,  though  most  often 
indirect,  of  all  geometrical  labours. 

This  general  proposition  being  fundamental,  since  it 
can  alone  give  our  definition  all  its  value,  it  is  indispen- 
sable to  enter  into  some  developments  upon  this  subject. 


STUDY    OF   THE    PROPERTIES   OF   FIGURES.  195 


PROPERTIES    OF    LINES   AND    SURFACES. 

When  we  examine  with  attention  the  geometrical  in- 
vestigations which  do  not  seem  to  relate  to  the  measure- 
ment of  extent,  we  find  that  they  consist  essentially  in 
the  study  of  the  different  properties  of  each  line  or  of  each 
'surface;  that  is,  in  the  knowledge  of  the  different  modes 
of  generation,  or  at  least  of  definition,  peculiar  to  each 
figure  considered.  Now  we  can  easily  establish  in  the 
most  general  manner  the  necessary  relation  of  such  a 
study  to  the  question  of  measurement,  for  which  the 
most  complete  knowledge  of  the  properties  of  each  form 
is  an  indispensable  preliminary.  This  is  concurrently 
proven  by  two  considerations,  equally  fundamental,  al- 
though quite  distinct  in  their  nature. 

NECESSITY  OF  THEIR  STUDY:  1.  To  find  the  most  suit- 
able Property.  The  first,  purely  scientific,  consists  in 
remarking  that,  if  we  did  not  know  any  other  character- 
istic property  of  each  line  or  surface  than  that  one  ac- 
cording to  which  geometers  had  first  conceived  it,  in 
most  cases  it  would  be  impossible  to  succeed  in  the  solu- 
tion of  questions  relating  to  its  measurement.  In  fact, 
it  is  easy  to  understand  that  the  different  definitions 
which  each  figure  admits  of  are  not  all  equally  suitable 
for  such  an  object,  and  that  they  even  present  the  most 
complete  oppositions  in  that  respect.  Besides,  since  the 
primitive  definition  of  each  figure  was  evidently  not  cho- 
sen with  this  condition  in  view,  it  is  clear  that  we  must 
not  expect,  in  general,  to  find  it  the  most  suitable  ; 
whence  results  the  necessity  of  discovering  others,  that 
is,  of  studying  as  far  as  is  possible  the  properties  of  the 
proposed  figure.  Let  us  suppose,  for  example,  that  the 


196 


GEOMETRY. 


circle  is  defined  to  be  "  the  curve  which,  with  the  same 
contour,  contains  the  greatest  area."  This  is  certainly 
a  very  characteristic  property,  but  we  would  evidently 
find  insurmountable  difficulties  in  trying  to  deduce  from 
such  a  starting  point  the  solution  of  the  fundamental 
questions  relating  to  the  rectification  or  to  the  quadra- 
ture of  this  curve.  It  is  clear,  in  advance,  that  the 
property  of  having  all  its  points  equally  distant  from  a 
fixed  point  must  evidently  be  much  better  adapted  to 
inquiries  of  this  nature,  even  though  it  be  not  precisely 
the  most  suitable.  In  like  manner,  would  Archimedes 
ever  have  been  able  to  discover  the  quadrature  of  the 
parabola  if  he  had  known  no  other  property  of  that  curve 
than  that  it  was  the  section  of  a  cone  with  a  circular 
base,  by  a  plane  parallel  to  its  generatrix  ?  The  pure- 
ly speculative  labours  of  preceding  geometers,  in  trans- 
forming this  first  definition,  were  evidently  indispensable 
preliminaries  to  the  direct  solution  of  such  a  question. 
The  same  is  true,  in  a  still  greater  degree,  with  respect 
to  surfaces.  To  form  a  just  idea  of  this,  we  need  only 
compare,  as  to  the  question  of  cubature  or  quadrature, 
the  common  definition  of  the  sphere  with  that  one,  no 
less  characteristic  certainly,  which  would  consist  in  re- 
garding a  spherical  body,  as  that  one  which,  with  the 
same  area,  contains  the  greatest  volume. 

No  more  examples  are  needed  to  show  the  necessity 
of  knowing,  so  far  as  is  possible,  all  the  properties  of  each 
line  or  of  each  surface,  in  order  to  facilitate  the  investi- 
gation of  rectifications,  of  quadratures,  and  of  cubatures, 
which  constitutes  the  final  object  of  geometry.  We  may 
even  say  that  the  principal  difficulty  of  questions  of  this 
kind  consists  in  employing  in  each  case  the  property  which 


STUDY    OF   THE    PROPERTIES   OF   FIGURES.  197 

• 

is  best  adapted  to  the  nature  of  the  proposed  problem. 
Thus,  while  we  continue  to  indicate,  for  more  precision, 
the  measurement  of  extension  as  the  general  destination 
of  geometry,  this  first  consideration,  which  goes  to  the 
very  bottom  of  the  subject,  shows  clearly  the  necessity 
of  including  in  it  the  study,  as  thorough  as  possible,  of 
the  different  generations  or  definitions  belonging  to  the 
same  form. 

2.  To  pass  from  the  Concrete  to  the  Abstract.  A 
second  consideration,  of  at  least  equal  importance,  con- 
sists in  such  a  study  being  indispensable  for  organizing 
in  a  rational  manner  the  relation  of  the  abstract  to  the 
concrete  in  geometry. 

The  science  of  geometry  having  to  consider  all  ima- 
ginable figures  which  admit  of  an  exact  definition,  it  ne- 
cessarily results  from  this,  as  we  have  remarked,  that 
questions  relating  to  any  figures  presented  by  nature 
are  always  implicitly  comprised  in  this  abstract  geome- 
try, supposed  to  have  attained  its  perfection.  But  when 
it  is  necessary  to  actually  pass  to  concrete  geometry,  we 
constantly  meet  with  a  fundamental  difficulty,  that  of 
knowing  to  which  of  the  different  abstract  types  we  are 
to  refer,  with  sufficient  approximation,  the  real  lines  or 
surfaces  which  we  have  to  study.  Now  it  is  for  the 
purpose  of  establishing  such  a  relation  that  it  is  particu- 
larly indispensable  to  know  the  greatest  possible  number 
of  properties  of  each  figure  considered  in  geometry. 

In  fact,  if  we  always  confined  ourselves  to  the  single 
primitive  definition  of  a  line  or  of  a  surface,  supposing 
even  that  we  could  then  measure  it  (which,  according  to 
the  first  order  of  considerations,  would  generally  be  im- 
possible), this  knowledge  would  remain  almost  necessa- 


198 


GEOMETRY. 


rily  barren  in  the  application,  since  we  should  not  ordi- 
narily know  how  to  recognize  that  figure  in  nature  when 
it  presented  itself  there  ;  to  ensure  that,  it  would  be  ne- 
cessary that  the  single  characteristic,  according  to  which 
geometers  had  conceived  it,  should  be  precisely  that  one 
whose  verification  external  circumstances  would  admit : 
a  coincidence  which  would  be  purely  fortuitous,  and  on 
which  we  could  not  count,  although  it  might  sometimes 
take  place.  It  is,  then,  only  by  multiplying  as  much  as 
possible  the  characteristic  properties  of  each  abstract  fig- 
ure, that  we  can  be  assured,  in  advance,  of  recognizing 
it  in  the  concrete  state,  and  of  thus  turning  to  account 
all  our  rational  labours,  by  verifying  in  each  case  the  defi- 
nition which  is  susceptible  of  being  directly  proven.  This 
definition  is  almost  always  the  only  one  in  given  cir- 
cumstances, and  varies,  on  the  other  hand,  for  the  same 
figure,  with  different  circumstances  ;  a  double  reason  for 
its  previous  determination. 

Illustration:  Orbits  of  the  Planets.  The  geometry 
of  the  heavens  furnishes  us  with  a  very  memorable  ex- 
ample in  this  matter,  well  suited  to  show  the  general  ne- 
cessity of  such  a  study.  We  know  that  the  ellipse  was 
discovered  by  Kepler  to  be  the  curve  which  the  planets 
describe  about  the  sun,  and  the  satellites  about  their 
planets.  Now  would  this  fundamental  discovery,  which 
re-created  astronomy,  ever  have  been  possible,  if  geom- 
eters had  been  always  confined  to  conceiving  the  el- 
lipse only  as  the  oblique  section  of  a  circular  cone  by  a 
plane?  No  such  definition,  it  is  evident,  would  admit 
of  such  a  verification.  The  most  general  property  of  the 
ellipse,  that  the  sum  of  the  distances  from  any  of  its  points 
to  two  fixed  points  is  a  constant  quantity,  is  undoubted- 


STUDY  OF  THE  PROPERTIES   OF  FIGURES.    ^99 

ly  much  more  susceptible,  by  its  nature,  of  causing  the 
curve  to  be  recognized  in  this  case,  but  still  is  not  di- 
rectly suitable.  The  only  characteristic  which  can  here 
be  immediately  verified  is  that  which  is  derived  from  the 
relation  which  exists  in  the  ellipse  between  the  length  of 
the  focal  distances  and  their  direction  ;  the  only  relation 
which  admits  of  an  astronomical  interpretation,  as  ex- 
pressing the  law  which  connects  the  distance  from  the 
planet  to  the  sun,  with  the  time  elapsed  since  the  begin- 
ning of  its  revolution.  It  was,  then,  necessary  that  the 
purely  speculative  labours  of  the  Greek  geometers  on  the 
properties  of  the  conic  sections  should  have  previously 
presented  their  generation  under  a  multitude  of  different 
points  of  view,  before  Kepler  could  thus  pass  from  the 
abstract  to  the  concrete,  in  choosing  from  among  all  these 
different  characteristics  that  one  which  could  be  most 
easily  proven  for  the  planetary  orbits. 

Illustration :  Figure  of  the  Earth.  Another  exam- 
ple of  the  same  order,  but  relating  to  surfaces,  occurs  in 
considering  the  important  question  of  the  figure  of  the 
earth.  If  we  had  never  known  any  other  property  of  the 
sphere  than  its  primitive  character  of  having  all  its  points 
equally  distant  from  an  interior  point,  how  would  we  ever 
have  been  able  to  discover  that  the  surface  of  the  earth 
was  spherical  ?  For  this,  it  was  necessary  previously  to 
deduce  from  this  definition  of  the  sphere  some  properties 
capable  of  being  verified  by  observations  made  upon  the 
surface  alone,  such  as  the  constant  ratio  which  exists  be- 
tween the  length  of  the  path  traversed  in  the  direction 
of  any  meridian  of  a  sphere  going  towards  a  pole,  and 
the  angular  height  of  this  pole  above  the  horizon  at  each 
point.  Another  example,  but  involving  a  much  longer 


200         •  GEOMETRY. 

series  of  preliminary  speculations,  is  the  subsequent  proof 
that  the  earth  is  not  rigorously  spherical,  but  that  its 
form  is  that  of  an  ellipsoid  of  revolution. 

After  such  examples,  it  would  be  needless  to  give  any 
others,  which  any  one  besides  may  easily  multiply.  All 
of  them  prove  that,  without  a  very  extended  knowledge 
of  the  different  properties  of  each  figure,  the  relation  of 
the  abstract  to  the  concrete,  in  geometry,  would  be  purely 
accidental,  and  that  the  science  would  consequently  want 
one  of  its  most  essential  foundations. 

Such,  then,  are  two  general  considerations  which  fully 
demonstrate  the  necessity  of  introducing  into  geometry  a 
great  number  of  investigations  which  have  not  the  meas- 
urement of  extension  for  their  direct  object ;  while  we 
continue,  however,  to  conceive  such  a  measurement  as 
being  the  final  destination  of  all  geometrical  science.  In 
this  way  we  can  retain  the  philosophical  advantages  of 
the  clearness  and  precision  of  this  definition,  and  still  in- 
clude in  it,  in  a  very  logical  though  indirect  manner,  all 
known  geometrical  researches,  in  considering  those  which 
do  not  seem  fo  relate  to  the  measurement  of  extension, 
as  intended  either  to  prepare  for  the  solution  of  the  final 
questions,  or  to  render  possible  the  application  of  the  so- 
lutions obtained. 

Having  thus  recognized,  as  a  general  principle,  the  close 
and  necessary  connexion  of  the  study  of  the  properties  of 
lines  and  surfaces  with  those  researches  which  constitute 
the  final  object  of  geometry,  it  is  evident  that  geometers, 
in  the  progress  of  their  labours,  must  by  no  means  con- 
strain themselves  to  keep  such  a  connexion  always  in 
view.  Knowing,  once  for  all,  how  important  it  is  to 
vary  as  much  as  possible  the  manner  of  conceiving  each 


STUDY   OF   THE    PROPERTIES   OF   FIGURES.  £01 

figure,  they  should  pursue  that  study,  without  consider- 
ing of  what  immediate  use  such  or  such  a  special  proper- 
ty may  be  for  rectifications,  quadratures,  and  cubatures. 
They  would  uselessly  fetter  their  inquiries  by  attaching 
a  puerile  importance  to  the  continued  establishment  of 
that  co-ordination. 

This  general  exposition  of  the  general  object  of  geom- 
etry is  so  much  the  more  indispensable,  since,  by  the  very 
nature  of  the  subject,  this  study  of  the  different  proper- 
ties of  each  line  and  of  each  surface  necessarily  composes 
by  far  the  greater  part  of  the  whole  body  of  geometrical 
researches.  .  Indeed,  the  questions  directly  relating  to  rec- 
tifications, to  quadratures,  and  to  cubatures,  are  evidently, 
by  themselves,  very  few  in  number  for  each  figure  con- 
sidered. On  the  other  hand,  the  study  of  the  properties 
of  the  same  figure  presents  an  unlimited  field  to  the  ac- 
tivity of  the  human  mind,  in  which  it  may  always  hope 
to  make  new  discoveries.  Thus,  although  geometers  have 
occupied  themselves  for  twenty  centuries,  with  more  or 
less  activity  undoubtedly,  but  without  any  real  interrup- 
tion, in  the  study  of  the  conic  sections,  they  are  far  from 
regarding  that  so  simple  subject  as  being  exhausted  ;  and 
it  is  certain,  indeed,  that  in  continuing  to  devote  them- 
selves to  it,  they  would  not  fail  to  find  still  unknown 
properties  of  those  different  -curves.  If  labours  of  this 
kind  have  slackened  considerably  for  a  century  past,  it 
is  not  because  they  are  completed,  but  only,  as  will  be 
presently  explained,  because  the  philosophical  revolution 
'in  geometry,  brought  about  by  Descartes,  has  singularly 
iiminished  the  importance  of  such  researches. 

It  results  from  the  preceding  considerations  that  not 
only  is  the  field  of  geometry  necessarily  infinite  because 


202 


GEOMETRY. 


of  the  variety  of  figures  to  be  considered,  but  also  in  vir- 
tue of  the  diversity  of  the  points  of  view  under  the  same 
figure  may  be  regarded.  This  last  conception  is,  indeed, 
that  which  gives  the  broadest  and  most  complete  idea  of 
the  whole  body  of  geometrical  researches.  We  see  that 
studies  of  this  kind  consist  essentially,  for  each  line  or  for 
each  surface,  in  connecting  all  the  geometrical  phenom- 
ena which  it  can  present,  with  a  single  fundamental  phe- 
nomenon, regarded  as  the  primitive  definition.  • 

THE   TWO  GENERAL  METHODS  OF   GEOMETRY. 

'  Having  now  explained  in  a  general  and  yet  precise 
manner  the  final  object  of  geometry,  and  shown  how  the 
science,  thus  defined,  comprehends  a  very  extensive  class 
of  researches  which  did  not  at  first  appear  necessarily  to 
belong  to  it,  there  remains  to  be  considered  the  method 
to  be  followed  for  the  formation  of  this  science.  This 
discussion  is  indispensable  to  complete  this  first  sketch 
of  the  philosophical  character  of  geometry.  I  shall  here 
confine  myself  to  indicating  the  most  general  considera- 
tion in  this  matter,  developing  and  summing  up  this  im- 
portant fundamental  idea  in  the  following  chapters. 

Geometrical  questions  may  be  treated  according  to 
two  methods  so  different,  that  there  result  from  them  two 
sorts  of  geometry,  so  to  say,  the  philosophical  character 
of  which  does  not  seem  to  me  to  have  yet  beei}  properly 
apprehended.  The  expressions  of  Synthetical  Geometry 
and  Analytical  Geometry,  habitually  employed  to  desig- 
nate them,  give  a  very  false  idea  of  them.  I  would  much 
prefer  the  purely  historical  denominations  of  Geometry  of 
the  Ancients  and  Geometry  of  the  Moderns,  which  have 
at  least  the  advantage  of  not  causing  their  true  charac- 


ITS  TWO  GENERAL  METHODS.     2(33 

ter  to  be  misunderstood.  But  I  propose  to  employ  hence- 
forth the  regular  expressions  of  Special  Geometry  and 
General  Geometry,  which  seem  to  me  suited  to  charac- 
terize with  precision  the  veritable  nature  of  the  two 
methods. 

Their  fundamental  Difference.  The  fundamental 
difference  between  the  manner  in  which  we  conceive 
Geometry  since  Descartes,  and  the  manner  in  which  the 
geometers  of  antiquity  treated  geometrical  questions,  is 
not  the  use  of  the  Calculus  (or  Algebra),  as  is  commonly 
thought  to  be  the  case.  On  the  one  hand,  it  is  certain 
that  the  use  of  the  calculus  was  not  entirely  unknown 
to  the  ancient  geometers,  since  they  used  to  make  con- 
tinual and  very  extensive  applications  of  the  theory  of 
proportions,  which  was  for  them,  as  a  means  of  deduc- 
tion, a  sort  of  real,  though  very  imperfect  and  especially 
extremely  limited  equivalent  for  our  present  algebra. 
The  calculus  may  even  be  employed  in  a  much  more 
complete  manner  than  they  have  used  it,  in  order  to  ob- 
tain certain  geometrical  solutions,  which  will  still  retain 
all'  the  essential  character  of  the  ancient  geometry  ;  this 
occurs  very  frequently  with  respect  to  those  problems  of 
geometry  of  two  or  of  three  dimensions,  which  are  com- 
monly designated  under  the  name  of  determinate.  On 
the  other  hand,  important  as  is  the  influence  of  the  cal- 
culus in  our  modern  geometry,  various  solutions  obtain- 
ed without  algebra  may  sometimes  manifest  the  peculiar 
character  which  distinguishes  it  from  the  ancient  geom- 
etry, although  analysis  is  generally  indispensable.  I  will' 
cite,  as  an  example,  the  method  of  Roberval  for  tangents, 
the  nature  of  which  is  essentially  modern,  and  which, 
however,  leads  in  certain  cases  to  complete  solutions, 


204 


GEOMETRY. 


without  any  aid  from  the  calculus.  It  is  not,  then,  the 
instrument  of  deduction  employed  which  is  the  principal 
distinction  between  the  two  courses  which  the  human 
mind  can  take  in  geometry. 

The  real  fundamental  difference,  as  yet  imperfectly 
apprehended,  seems  to  me  to  consist  in  the  very  nature 
of  the  questions  considered.  In  truth,  geometry,  view- 
ed as  a  whole,  and  supposed  to  have  attained  entire  per- 
fection, must,  as  we  have  seen  on  the  one  hand,  em- 
brace all  imaginable  figures,  and,  on  the  other,  discover 
all  the  properties  of  each  figure.  It  admits,  from  this 
double  consideration,  of  being  treated  according  to  two 
essentially  distinct  plans ;  either,  1°,  by  grouping  to- 
gether all  the  questions,  however  different  they  may  be, 
which  relate  to  the  same  figure,  and  isolating  those  re- 
lating to  different  bodies,  whatever  analogy  there  may 
exist  between  them  ;  or,  2°,  on  the  contrary,  by  uniting 
under  one  point  of  view  all  similar  inquiries,  to  whatever 
different  figures  they  may  relate,  and  separating  the 
questions  relating  to  the  really  different  properties  of  the 
same  body.  In  a  word,  the  whole  body  of  geometry 
may  be  essentially  arranged  either  with  reference  to  the 
bodies  studied  or  to  the  phenomena  to  be  considered. 
The  first  plan,  which  is  the  most  natural,  was  that  of 
the  ancients  ;  the  second,  infinitely  more  rational,  is  that 
of  the  moderns  since  Descartes. 

Geometry  of  the  Ancients.  Indeed,  the  principal  char- 
acteristics of  the  ancient  geometry  is  that  they  studied, 
one  by  one,  the  different  lines  and  the  different  surfaces, 
not  passing  to  the  examination  of  a  new  figure  till  they 
thought  they  had  exhausted  all  that  there  was  interest- v 
ing  in  the  figures  already  known.  In  this  way  of  pro- 


THE   MODERN   GEOMETRY.  205 

Deeding,  when  they  undertook  the  study  of  a  new  curve, 
the  whole  of  the  labour  bestowed  on  the  preceding  ones 
could  not  offer  directly  any  essential  assistance,  other- 
wise than  by  the  geometrical  practice  to  which  it  had 
trained  the  mind.  Whatever  might  be  the  real  similari- 
ty of  the  questions  proposed  as  to  two  different  figures, 
the  complete  knowledge  acquired  for  the  one  could  not 
at  all  dispense  with  taking  up  again  the  whole  of  the  in- 
vestigation for  the  other.  Thus  the  progress  of  the  mind 
was  never  assured  ;  so  that  they  could  not  be  certain,  in 
advance,  of  obtaining  any  solution  whatever,  however 
analogous  the  proposed  problem  might  be  to  questions 
which  had  been  already  resolved.  Thus,  for  example, 
the  determination  of  the  tangents  to  the  three  conic  sec- 
tions did  not  furnish  any  rational  assistance  for  drawing 
the  tangent  to  any  other  new  curve,  such  as  the  con- 
choid, the  cissoid,  &c.  In  a  word,  the  geometry  of.  the 
ancients  was,  according  to  the  expression  proposed  above, 
essentially  special. 

Geometry  of  the  Moderns.  In  the  system  of  the 
moderns,  geometry  is,  on  the  contrary,  eminently  gen- 
eral, that  is  to  say,  relating  to  any  figures  whatever.  It 
is  easy  to  understand,  in  the  first  place,  that  all  geomet- 
rical expressions  of  any  interest  may  be  proposed  with 
reference  to  all  imaginable  figures.  This  is  seen  direct- 

o  o 

ly  in  the  fundamental  problems — of  rectifications,  quad- 
ratures, and  cubatures — which  constitute,  as  has  been 
shown,  the  final  object  of  geometry.  But  this  remark 
is  no  less  incontestable,  even  for  investigations  which  re- 
late to  the  different  properties  of  lines  and  of  surfaces, 
and  of  which  the  most  essential,  such  as  the  question  of 
tangents  or  of  tangent  planes,  the  theory  of  curvatures, 


206 


GEOMETRY. 


&c.,  are  evidently  common  to  all  figures  whatever.  The 
very  few  investigations  which  are  truly  peculiar  to  par- 
ticular figures  have  only  an  extremely  secondary  im- 
portance. This  being  understood,  modern  geometry  con- 
sists essentially  in  abstracting,  in  order  to  treat  it  by  it- 
self, in  an  entirely  general  manner,  every  question  re- 
lating to  the  same  geometrical  phenomenon,  in  whatever 
bodies  it  may  be  considered.  The  application  of  the 
universal  th'eories  thus  constructed  to  the  special  deter- 
mination of  the  phenomenon  which  is  treated  of  in  each 
particular  body,  is  now  regarded  as  only  a  subaltern  la- 
bour, to  be  executed  according  to  invariable  rules,  and 
the  success  of  which  is  certain  in  advance.  This  labour 
is,  in  a  word,  of  the  same  character  as  the  numerical  cal- 
culation of  an  analytical  formula.  There  can  be  no  other 
merit  in  it  than  that  of  presenting  in  each  case  the  so- 
lution which  is  necessarily  furnished  by  the  general 
method,  with  all  the  simplicity  and  elegance  which  the 
line  or  the  surface  considered  can  admit  of.  But  no  real 
importance  is  attached  to  any  thing  but  the  conception 
and  the  complete  solution  of  a  new  question  belonging 
to  any  figure  whatever.  Labours  of  this  kind  are  alone 
regarded  as  producing  any  real  advance  in  science.  The 
attentibn  of  geometers,  thus  relieved  from  the  examina- 
tion of  the  peculiarities  of  different  figures,  and  wholly 
directed  towards  general  questions,  has  been  thereby  able 
to  elevate  itself  to  the  consideration  of  new  geometrical 
conceptions,  which,  applied  to  the  curves  studied  by  the 
ancients,  have  led  to  the  discovery  of  important  proper- 
ties which  they  had  not  before  even  suspected.  Such  is 
geometry,  since  the  radical  revolution  produced  by  Des- 
cartes in  the  general  system  of  the  science. 


SUPERIORITY   OF   THE   MODERN   METHOD.  £07 

The  Superiority  of  the  modern  Geometry,  The  mere 
indication  of  the  fundamental  character  of  each  of  the 
two  geometries  is  undoubtedly  sufficient  to  make  appa- 
rent the  immense  necessary  superiority  of  modern  geom- 
etry. We  may  even  say  that,  before  the  great  concep- 
tion of  Descartes,  rational  geometry  was  not  truly  con- 
stituted upon  definitive  bases,  whether  in  its  abstract  or 
concrete  relations.  In  fact,  as  regards  science,  consid- 
ered speculatively,  it  is  clear  that,  in  continuing  indefi- 
nitely to  follow  the  course  of  the  ancients,  as  did  the 
moderns  before  Descartes,  and  even  for  a  little  while  af- 
terwards, by  adding  some  new  curves  to  the  small  num- 
ber of  those  which  they  had  studied,  the  progress  thus 
made,  however  rapid  it  might  have  been,  would  still  be 
found,  after  a  long  series  of  ages,  to  be  very  inconsider- 
able in  comparison  with  the  general  system  of  geometry, 
seeing  the  infinite  variety  of  the  forms  which  would  still 
have  remained  to  be  studied.  On  the  contrary,  at  each 
question  resolved  according  to  the  method  of  the  mod- 
erns, the  number  of  geometrical  problems  to  be  resolved 
is  then,  once  for  all,  diminished  by  so  much  with  respect 
to  all  possible  bodies.  *Another  consideration  is,  that  it 
resulted,  from  their  complete  want  of  general  methods, 
that  the  ancient  geometers,  in  all  their  investigations, 
were  entirely  abandoned  to  their  own  strength,  without 
ever  having  the  certainty  of  obtaining,  sooner  or  later, 
any  solution  whatever.  Though  this  imperfection  of  the 
-cience  was  eminently  suited  to  call  forth  all  their  ad- 
mirable sagacity,  it  necessarily  rendered  their  progress 
extremely  slow ;  we  can  form  some  idea  of  this  by  the 
considerable  time  which  they  employed  in  the  study  of 
the  conic  sections.  Modern  geometry,  making  the  prog- 


208 


GEOMETRY. 


ress  of  our  mind  certain,  permits  us,  on  the  contrary,  to 
make  the  greatest  possible  use  of  the  forces  of  our  intel- 
ligence, which  the  ancients  were  often  obliged  to  waste 
on  very  unimportant  questions. 

A  no  less  important  difference  between  the  two  sys- 
tems appears  when  we  come  to  consider  geometry  in  the 
concrete  point  of  view.  Indeed,  we  have  already  re- 
marked that  the  relation  of  the  abstract  to  the  concrete 
in  geometry  can  be  founded  upon  rational  bases  only  so 
far  as  the  investigations  are  made  to  bear  directly  upon 
all  imaginable  figures.  In  studying  lines,  only  one  by 
one,  whatever  may  be,  the  number,  always  necessarily 
very  small,  of  those  which  we  shall  have  considered, "the 
application  of  such  theories  to  figures  really  existing  in 
nature  will  never  have  any  other  than  an  essentially 
accidental  character,  since  there  is  nothing  to  assure  us 
that  these  figures  can  really  be  brought  under  the  ab- 
stract types  considered  by  geometers: 

Thus,  for  example,  there  is  certainly  something  for- 
tuitous in  the  happy  relation  established  between  the 
speculations  of  the  Greek  geometers  upon  the  conic  sec- 
tions and  the  determination  of  the  true  planetary  orbits. 
In  continuing  geometrical  researches  upon  the  same  plan, 
there  was  no  good  reason  for  hoping  for  similar  coinci- 
dences ;  and  it  would  have  been  possible,  in  these  spe- 
cial studies,  that  the  researches  of  geometers  should  have 
been  directed  to  abstract  figures  entirely  incapable  of  any 
application,  while  they  neglected  others,  susceptible  per- 
haps of  an  important  and  immediate  application.  It  is 
clear,  at  least,  that  nothing  positively  guaranteed  the 
necessary  applicability  of  geometrical  speculations.  It 
is  quite  another  thing  in  the  modern  geometry.  From 


THE   USE    OF    THE   ANCIENT    GEOMETRY.     £09 

the  single  circumstance  that  in  it  we  proceed  by  general 
questions  relating  to  any  figures  whatever,  we  have  in 
advance  the  evident  certainty  that  the  figures  really  ex- 
isting in  the  external  world  could  in  no  case  escape  the 
appropriate  theory  if  the  geometrical  phenomenon  which 
it  considers  presents  itself  in  them. 

From  these  different  considerations,  we  see  that  the 
ancient  system  of  geometry  wears  essentially  the  char- 
acter of  the  infancy  of  the  science,  which  did  not  begin 
to  become  completely  rational  till  after  the  philosophical 
resolution  produced  by  Descartes.  But  it  is  evident,  on 
the  other  hand,  that  geometry  could  not  be  at  first  con- 
ceived except  in  this  special  manner.  General  geome- 
try would  not  have  been  possible,  and  its  necessity  could 
not  even  have  been  felt,  if  a  long  series  of  special  labours 
on  the  most  simple  figures  had  not  previously  furnished 
bases  for  the  conception  of  Descartes,  and  rendered  ap- 
parent the  impossibility  of  persisting  indefinitely  in  the 
primitive  geometrical  philosophy. 

The  Ancient  the  Base  of  the  Modern.  From  this  last 
consideration  we  must  infer  that,  although  the  geometry 
which  I  have  called  general  must  be  now  regarded  as 
the  only  true  dogmatical  geometry,  and  that  to  which 
we  shall  chiefly  confine  ourselves,  the  other  having  no 
longer  much  more  than  an  historical  interest,  nevertheless 
it  is  not  possible  to  entirely  dispense  with  special  geom- 
etry in  a  rational  exposition  of  the  science.  We  un- 
doubtedly need  not  borrow  directly  from  ancient  geom- 
etry all  the  results  which  it  has  furnished  ;  but,  from  the 
very  nature  of  the  subject,  it  is  necessarily  impossible  en- 
tirely to  dispense  with  the  ancient  method,  which  will 
always  serve  as  the  preliminary  basis  of  the  science,  dog. 

O 


210  GEOMETRY. 

matically  as  well  as  historically.  The  reason  of  this  is 
easy  to  understand.  In  fact,  general  geometry  being 
essentially  founded,  as  we  shall  soon  establish,  upon  tbe 
employment  of  the  calculus  in  the  transformation  of  geo- 
metrical into  analytical  considerations,  such  a  manner  of 
proceeding  could  not  take  possession  of  the  subject  im- 
mediately at  its  origin.  We  know  that  the  application 
of  mathematical  analysis,  from  its  nature,  can  never  com- 
mence any  science  whatever,  since  evidently  it  cannot 
be  employed  until  the  science  has  already  been  sufficient- 
ly cultivated  to  establish,  with  respect  to  the  phenomena 
considered,  some  equations  which  can  serve  as  starting 
points  for  the  analytical  operations.  These  fundamental 
equations  being  once  discovered,  analysis  will  enable  us 
to  deduce  from  them  a  multitude  of  consequences  which 
it  would  have  been  previously  impossible  even  to  sus- 
pect ;  it  will  perfect  the  science  to  an  immense  degree, 
both  with  respect  to  the  generality  of  its  conceptions  and 
to  the  complete  co-ordination  established  between  them. 
But  mere  mathematical  analysis  could  never  be  sufficient 
to  form  the  bases  of  any  natural  science,  not  even  to  de- 
monstrate them  anew  when  they  have  once  been  estab- 
lished. Nothing  can  dispense  with  the  direct  study  of 
the  subject,  pursued  up  to  the  point  of  the  discovery  of 
precise  relations. 

We  thus  see  that  the  geometry  of  the  ancients  will 
always  have,  by  its  nature,  a  primary  part,  absolutely  ne- 
cessary and  more  or  less  extensive,  in  the  complete  sys- 
tem of  geometrical  knowledge.  It  forms  a  rigorously 
indispensable  introduction  to  general  geometry.  But  it 
is  to  this  that  it  must  be  limited  in  a  completely  dog- 
matic exposition.  I  will  consider,  then,  directly,  in  the 


THE  USE  OF  THE  ANCIENT  GEOMETRY.  211 

following  chapter,  this  special  or  preliminary  geometry 
restricted  to  exactly  its  necessary  limits,  in  order  to  oc- 
cupy myself  thenceforth  only  with  the  philosophical  ex- 
amination of  general  or  definitive  geometry,  the  only  one 
which  is  truly  rational,  and  which  at  present  essentially 
composes  the  science. 


CHAPTER  II. 

ANCIENT  OR  SYNTHETIC  GEOMETRY. 

THE  geometrical  method  of  the  ancients  necessarily 
constituting  a  preliminary  department  in  the  dogmatical 
system  of  geometry,  designed  to  furnish  general  geome- 
try with  indispensable  foundations,  it  is  now  proper  to 
begin  with  determining  wherein  strictly  consists  this  pre- 
liminary function  of  special  geometry,  thus  reduced  to 
the  narrowest  possible  limits. 


ITS  PROPER  EXTENT. 


Lines  ;  Polygons  ;  Polyhedrons.  In  considering  it 
under  this  point  of  view,  it  is  easy  to  recognize  that  we 
might  restrict  it  to  the  study  of  the  right  line  alone  for 
what  concerns  the  geometry  of  lines ;  to  the  quadrature 
of  rectilinear  plane  areas ;  and,  lastly,  to  the  cubature  of 
bodies  terminated  by  plane  faces.  The  elementary  prop- 
ositions relating  to  these  three  fundamental  questions 
form,  in  fact,  the  necessary  starting  point  of  all  geomet- 
rical inquiries  ;  they  alone  cannot  be  obtained  except  by 
a  direct  study  of  the  subject ;  while,  on  the  contrary, 
the  complete  theory  of  all  other  figures,  even  that  of  the 
circle,  and  of  the  surfaces  and  volumes  which  are  con- 
nected with  it,  may  at  the  present  day  be  completely 
comprehended  in  the  domain  of  general  or  analytical 
geometry ;  these  primitive  elements  at  once  furnishing 
equations  which  are  sufficient  to  allow  of  the  application 


ITS   PROPER   EXTENT.  £13 

of  the  calculus  to  geometrical  questions,  which  would  not 
have  been  possible  without  this  previous  condition. 

It  results  from  this  consideration  that,  in  common  prac- 
tice, we  give  to  elementary  geometry  more  extent  than 
would  be  rigorously  necessary  to  it ;  since,  besides  the 
right  line,  polygons,  and  polyhedrons,  we  also  include  in 
it  the  circle  and  the  "  round"  bodies  ;  the  study  of  which 
might,  however,  be  as  purely  analytical  as  that,  for  ex- 
ample, of  the  conic  sections.  An  unreflecting  veneration 
for  antiquity  contributes  to  maintain  this  defect  in  meth- 
od ;  but  the  best  reason  which  can  be  given  for  it  is  the 
serious  inconvenience  for  ordinary  instruction  which  there 
would  be  in  postponing,  to  so  distant  an  epoch  of  mathe- 
matical education,  the  solution  of  several  essential  ques- 
tions, which  are  susceptible  of  a  direct  and  continual  ap- 
plication to  a  great  number  of  important  uses.  In  fact, 
to  proceed  in  the  most  rational  manner,  we  should  em- 
ploy the  integral  calculus  in  obtaining  the  interesting 
results  relating  to  the  length  or  the  area  of  the  circle,  or 
to  the  quadrature  of  the  sphere,  &c.,  which  have  been 
determined  by  the  ancients  from  extremely  simple  con- 
siderations. This  inconvenience  would  be  of  little  im- 
portance with  regard  to  the  persons  destined  to  study 
the  whole  of  mathematical  science,  and  the  advantage 
of  proceeding  in  a  perfectly  logical  order  would  have  a 
much  greater  comparative  value.  But  the  contrary  case 
being  the  more  frequent,  theories  so  essential  have  neces- 
sarily been  retained  in  elementary  geometry.  Perhaps 
the  conic  sections,  the  cycloid,  &c.,  might  be  advanta- 
geously added  in  such  cases. 

Not  to  be  farther  restricted.  While  this  preliminary 
portion  of  geometry,  which  cannot  be  founded  on  the  ap- 


214  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

plication  of  the  calculus,  is  reduced  by  its  nature  to  a 
very  limited  series  of  fundamental  researches,  relating  to 
the  right  line,  polygonal  areas,  and  polyhedrons,  it  is  cer- 
tain, on  the  other  hand,  that  we  cannot  restrict  it  any 
more ;  although,  by  a  veritable  abuse  of  the  spirit  of 
analysis,  it  has  been  recently  attempted  to  present  the 
establishment  of  the  principal  theorems  of  elementary  ge- 
ometry under  an  algebraical  point  of  view.  Thus  some 
have  pretended  to  demonstrate,  by  simple  abstract  con- 
siderations of  mathematical  analysis,  the  constant  rela- 
tion which  exists  between  the  three  angles  of  a  rectilin- 
ear triangle,  the  fundamental  proposition  of  the  theory 
of  similar  triangles,  that  of  parallelopipedons,  &c.  ;  in  a 
word,  precisely  the  only  geometrical  propositions  which 
cannot  be  obtained  except  by  a  direct  study  of  the  sub- 
ject, without  the  calculus  being  susceptible  of  having 
any  part  in  it.  Such  aberrations  are  the  unreflecting 
exaggerations  of  that  natural  and  philosophical  tendency 
which  leads  us  to  extend  farther  and  farther  the  influ- 
ence of  analysis  in  mathematical  studies.  In  mechan- 
ics, the  pretended  analytical  demonstrations  of  the  paral- 
lelogram of  forces  are  of  similar  character. 

The  viciousness  of  such  a  manner  of  proceeding  follows 
from  the  principles  previously  presented.  We  have  al- 
ready, in  fact,  recognized  that,  since  the  calculus  is  not, 
and  cannot  be,  any  thing  but  a  means  of  deduction,  it 
would  indicate  a  radically  false  idea  of  it  to  wish  to 
employ  it  in  establishing  the  elementary  foundations  of 
any  science  whatever  ;  for  on  what  would  the  analytical 
reasonings  in  such  an  operation  repose  ?  A  labour  of  this 
nature,  very  far  from  really  perfecting  the  philosophical 
character  of  a  science,  would  constitute  a  return  towards 


ITS    PROPER  EXTENT.  215 

the  metaphysical  age,  in  presenting  real  facts  as  mere 
logical  abstractions. 

When  we  examine  in  themselves  these  pretended  an- 
alytical demonstrations  of  the  fundamental  propositions 
of  elementary  geometry,  we  easily  verify  their  necessary 
want  of  meaning.  They  are  all  founded  on  a  vicious 
manner  of  conceiving  the  principle  of  homogeneity,  the 
true  general  idea  of  which  was  explained  in  the  second 
3hapter  of  the  preceding  book.  These  demonstrations 
suppose  that  this  principle  does  not  allow  us  to  admit  the 
coexistence  in  the  same  equation  of  numbers  obtained  by 
different  concrete  comparisons,  which  is  evidently  false, 
and  contrary  to  the  constant  practice  of  geometers.  Thus 
it  is  easy  to  recognize  that,  by  employing  the  law  of  ho- 
mogeneity in  this  arbitrary  and  illegitimate  acceptation, 
we  could  succeed  in  "demonstrating,"  with  quite  as  much 
apparent  rigour,  propositions  whose  absurdity  is  manifest 
at  the  first  glance.  In  examining  attentively,  for  ex- 
ample, the  procedure  by  the  aid  of  which  it  has  been  at- 
tempted to  prove  analytically  that  the  sum  of  the  three 
angles  of  any  rectilinear  triangle  is  constantly  equal  to 
two  right  angles,  we  see  that  it  is  founded  on  this  pre- 
liminary principle  that,  if  two  triangles  have  two  of  their 
angles  respectively  equal,  the  third  angle  of  the  one  will 
necessarily  be  equal  to  the  third  angle  of  the  other.  This 
first  point  being  granted,  the  proposed  relation  is  imme- 
diately deduced  from  it  in  a  very  exact  and  simple  man- 
ner. Now  the  analytical  consideration  by  which  this 
previous  proposition  has  been  attempted  to  be  establish- 
ed, is  of  such  a  nature  that,  if  it  could  be  correct,  we 
could  rigorously  deduce  from  it,  in  reproducing  it  con- 
versely, this  palpable  absurdity,  that  two  sides  of  a  tri- 


216  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

angle  are  sufficient,  without  any  angle,  for  the  entire  de- 
termination of  the  third  side.  We  may  make  analogous 
remarks  on  all  the  demonstrations  of  this  sort,  the  soph- 
isms of  which  will  be  thus  verified  in  a  perfectly  appa- 
rent manner. 

The  more  reason  that  we  have  here  to  consider  geome- 
try as  being  at  the  present  day  essentially  analytical,  the 
more  necessary  was  it  to  guard  against  this  abusive  ex- 
aggeration of  mathematical  analysis,  according  to  which 
all  geometrical  observation  would  be  dispensed  with,  in 
establishing  upon  pure  algebraical  abstractions  the  very 
foundations  of  this  natural  science. 

Attempted  Demonstrations  of  Axioms,  8{c.  Another 
indication  that  geometers  have  too  much  overlooked  the 
character  of  a  natural  science  which  is  necessarily  inhe- 
rent in  geometry,  appears  from  their  vain  attempts,  so 
long  made,  to  demonstrate  rigorously,  not  by  the  aid  of 
the  calculus,  but  by  means  of  certain  constructions,  sev- 
eral fundamental  propositions  of  elementary  geometry. 
Whatever  may  be  effected,  it  will  evidently  be  impossi- 
ble to  avoid  sometimes  recurring  to  simple  and  direct  ob- 
servation in  geometry  as  a  means  of  establishing  va- 
rious results.  While,  in  this  science,  the  phenomena 
which  are  considered  are,  by  virtue  of  their  extreme  sim- 
plicity, much  more  closely  connected  with  one  another 
than  those  relating  to  any  other  physical  science,  some 
must  still  be  found  which  cannot  be  deduced,  and  which, 
on  the  contrary,  serve  as  starting  points.  It  may  be 
admitted  that  the  greatest  logical  perfection  of  the  sci- 
ence is  to  reduce  these  to  the  smallest  number  possible, 
but  it  would  be  absurd  to  pretend  to  make  them  com- 
pletely disappear.  I  avow,  moreover,  that  I  find  fewer 


GEOMETRY    OF  THE  RIGHT   LINE.         £17 

real  inconveniences  in  extending,  a  little  beyond  what 
would  be  strictly  necessary,  the  number  of  these  geo- 
metrical notions  thus  established  by  direct  observation, 
provided  they  are  sufficiently  simple,  than  in  making 
them  the  subjects  of  complicated  and  indirect  demonstra- 
tions, even  when  these  demonstrations  may  be  logically 
irreproachable. 

The  true  dogmatic  destination  of  the  geometry  of  the 
ancients,  reduced  to  its  least  possible  indispensable  de- 
velopments, having  thus  been  characterized  as  exactly  as 
possible,  it  is  proper  to  consider  summarily  each  of  the 
principal  parts  of  which  it  must  be  composed.  I  think 
that  I  may  here  limit  myself  to  considering  the  first  and 
the  most  extensive  of  these  parts,  that  which  has  for  its 
object  the  study  of  the  right  line  ;  the  two  other  sections, 
namely,  the  quadrature  of  polygons  and  the  cubature 
of  polyhedrons,  from  their  limited  extent,  not  being  ca- 
pable of  giving  rise  to  any  philosophical  consideration  of 
any  importance,  distinct  from  those  indicated  in  the  pre- 
ceding chapter  with  respect  to  the  measure  of  areas  and 
of  volumes  in  general. 

GEOMETRY   OF    THE    RIGHT   LINE. 

The  final  question  which  we  always  have  in  view  in 
the  study  of  the  right  line,  properly  consists  in  deter- 
mining, by  means  of  one  another,  the  different  elements 
of  any  right-lined  figure  whatever ;  which  enables  us 
always  to  know  indirectly  the  length  and  position  of  a 
right  line,  in  whatever  circumstances  it  may  be  placed. 
This  fundamental  problem  is  susceptible  of  two  general 
solutions,  the  nature  of  which  is  quite  distinct,  the  one 
graphical,  the  other  algebraic.  The  first,  though  very 


218   ANCIENT    OR  SYNTHETIC    GEOMETRY. 

imperfect,  is  that  which  must  be  first  considered,  be- 
cause it  is  spontaneously  derived  from  the  direct  study 
of  the  subject ;  the  second,  much  more  perfect  in  the 
most  important  respects,  cannot  be  studied  till  after- 
wards, because  it  is  founded  upon  the  previous  knowl- 
edge of  the  other. 

GRAPHICAL    SOLUTIONS. 

The  graphical  solution  consists  in  constructing  at  will 
the  proposed  figure,  either  with  the  same  dimensions,  or, 
more  usually,  with  dimensions  changed  in  any  ratio  what- 
ever. The  first  mode  need  merely  be  mentioned  as  be- 
ing the  most  simple  and  the  one  which  would  first  occur 
to  the  rnind,  for  it  is  evidently,  by  its  nature,  almost  en- 
tirely incapable  of  application.  The  second  is,  on  the 
contrary,  susceptible  of  being  most  extensively  and  most 
usefully  applied.  We  still  make  an  important  and  con- 
tinual use  of  it  at  the  present  day,  not  only  to  represent 
with  exactness  the  forms  of  bodies  and  their  relative  po- 
sitions, but  even  for  the  actual  determination  of  geomet- 
rical magnitudes,  when  we  do  not  need  great  precision. 
The  ancients,  in  consequence  of  the  imperfection  of  their 
geometrical  knowledge,  employed  this  procedure  in  a 
much  more  extensive  manner,  since  it  was  for  a  long  time 
the  only  one  which  they  could  apply,  even  in  the  most 
important  precise  determinations.  It  was  thus,  for  exam- 
ple, that  Aristarchus  of  Samos  estimated  the  relative  dis- 
tance from  the  sun  and  from  the  moon  to  the  earth,  by 
making  measurements  on  a  triangle  constructed  as  ex- 
actly as  possible,  so  as  to  be  similar  to  the  right-angled 
triangle  formed  by  the  three  bodies  at  the  instant  when 
the  moon  is  in  quadrature,  and  when  an  observation  of 


GRAPHICAL  SOLUTIONS.        219 

the  angle  at  the  earth  would  consequently  be  sufficient  to 
define  the  triangle.  Archimedes  himself,  although  he  was 
the  first  to  introduce  calculated  determinations  into  .ge- 
ometry, several  times  employed  similar  means.  The 
formation  of  trigonometry  did  not  cause  this  method  to 
be  entirely  abandoned,  although  it  greatly  diminished  its 
use  ;  the  Greeks  and  the  Arabians  continued  to  employ 
it  for  a  great  number  of  researches,  in  which  we  now  re- 
gard the  use  of  the  calculus  as  indispensable. 

This  exact  reproduction  of  any  figure  whatever  on  a 
different  scale  cannot  present  any  great  theoretical  diffi- 
culty when  all  the  parts  of  the  proposed  figure  lie  in  the 
same  plane.  But  if  we  suppose,  as  most  frequently  hap- 
pens, that  they  are  situated  in  different  planes,  we  see, 
then,  a  new  order  of  geometrical  considerations  arise. 
The  artificial  figure,  which  is  constantly  plane,  not  being 
capable,  in  that  case,  of  being  a  perfectly  faithful  image 
of  the  real  figure,  it  is  necessary  previously  to  fix  with 
precision  the  mode  of  representation,  which  gives  rise  to 
different  systems  of  Projection. 

It  then  remains  to  be  determined  according  to  what 
laws  the  geometrical  phenomena  correspond  in  the  two 
figures.  This  consideration  generates  a  new  series  of 
geometrical  investigations,  the  final  object  of  which  is 
properly  to  discover  how  we  can  replace  constructions  in 
relief  by  plane  constructions.  The  ancients  had  to  re- 
solve several  elementary  questions  of  this  kind  for  vari- 
ous cases  in  which  we  now  employ  spherical  trigonome- 
try, principally  for  different  problems  relating  to  the  ce- 
lestial sphere.  Such  was  the  object  of  their  analemmas, 
and  of  the  other  plane  figures  which  for  a  long  time  sup- 
plied the  place  of  the  calculus.  We  see  by  this  that  the 


220  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

ancients  really  knew  the  elements  of  what  we  now  name 
Descriptive  Geometry,  although  they  did  not  conceive  it 
in  a  distinct  and  general  manner. 

I  think  it  proper  briefly  to  indicate  in  this  place  the 
true  philosophical  character  of  this  "Descriptive  Geome- 
try ;"  although,  being  essentially  a  science  of  application, 
it  ought  not  to  be  included  within  the  proper  domain  of 
this  work. 

DESCRIPTIVE     GEOMETRY. 

All  questions  of  geometry  of  three  dimensions  neces- 
sarily give  rise,  when  we  consider  their  graphical  solu- 
tion, to  a  common  difficulty  which  is  peculiar  to  them ; 
that  of  substituting  for  the  different  constructions  in  re- 
lief, which  are  necessary  to  resolve  them  directly,  and 
which  it  is  almost  always  impossible  to  execute,  simple 
equivalent  plane  constructions,  by  means  of  which  we 
finally  obtain  the  same  results.  Without  this  indispen- 
sable transformation,  every  solution  of  this  kind  would  be 
evidently  incomplete  and  really  inapplicable  in  practice, 
although  theoretically  the  constructions  in  space  are  usu- 
ally preferable  as  being  more  direct.  It  was  in  order  to 
furnish  general  means  for  always  effecting  such  a  trans- 
formation that  Descriptive  Geometry  was  created,  and 
formed  into  a  distinct  and  homogeneous  system,  by  the 
illustrious  MONGE.  He  invented,  in  the  first  place,  a  uni- 
form method  of  representing  bodies  by  figures  traced  on  a 
single  plane,  by  the  aid  of  projections  on  two  different 
planes,  usually  perpendicular  to  each  other,  and  one  of 
which  is  supposed  to  turn  about  their  common  intersec- 
tion so  as  to  coincide  with  the  other  produced ;  in  this 
system,  or  in  any  other  equivalent  to  it,  it  is  sufficient 


DESCRIPTIVE    GEOMETRY.  £21 

to  regard  points  and  lines  as  being  determined  by  their 
projections,  and  surfaces  by  the  projections  of  their  gen- 
erating lines.  This  being  established,  Monge — analyz- 
ing with  profound  sagacity  the  various  partial  labours  of 
this  kind  which  had  before  been  executed  by  a  number 
of  inconguous  procedures,  and  considering  also,  in  a  gen- 
eral and  direct  manner,  in  what  any  questions  of  that 
nature  must  consist — found  that  they  could  always  be 
reduced  to  a  very  small  number  of  invariable  abstract 
problems,  capable  of  being  resolved  separately,  once  for 
all,  by  uniform  operations,  relating  essentially  some  to 
the  contacts  and  others  to  the  intersections  of  surfaces. 
Simple  and  entirely  general  methods  for  the  graphical 
solution  of  these  two  orders  of  problems  having  been 
formed,  all  the  geometrical  questions  which  may  arise  in 
any  of  the  various  arts  of  construction — stone-cutting, 
carpentry,  perspective,  dialling,  fortification,  &c. — can 
henceforth  be  treated  as  simple  particular  cases  of  a  sin- 
gle theory,  the  invariable  application  of  which  will  al- 
ways necessarily  lead  to  an  exact  solution,  which  may 
be  facilitated  in  practice  by  profiting  by  the  peculiar 
circumstances  of  each  case. 

This  important  creation  deserves  in  a  remarkable  de- 
gree to  fix  the  attention  of  those  philosophers  who  con- 
sider all  that  the  human  species  has  yet  effected  as  a 
first  step,  and  thus  far  the  only  really  complete  one,  to- 
wards that  general  renovation  of  human  labours,  which 
must  imprint  upon  all  our  arts  a  character  of  precision 
and  of  rationality,  so  necessary  to  their  future  progress 
Such  a  revolution  must,  in  fact,  inevitably  commence 
with  that  class  of  industrial  labours,  which  is  essentially 


222  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

connected  with  that  science  which  is  the  most  simple, 
the  most  perfect,  and  the  most  ancient.  It  cannot  fail 
to  extend  hereafter,  though  with  less  facility,  to  all  other 
practical  operations.  Indeed  Monge  himself,  who  con- 
ceived the  true  philosophy  of  the  arts  better  than  any  one 
else,  endeavoured  to  sketch  out  a  corresponding  system 
for  the  mechanical  arts. 

Essential  as  the  conception  of  descriptive  geometry 
really  is,  it  is  very  important  not  to  deceive  ourselves 
with  respect  to  its  true  destination,  as  did  those  who, 
in  the  excitement  of  its  first  discovery,  saw  in  it  a  means 
of  enlarging  the  general  and  abstract  domain  of  rational 
geometry.  The  result  has  in  no  way  answered  to  these 
mistaken  hopes.  And,  indeed,  is  it  not  evident  that  de- 
scriptive geometry  has  no  special  value  except  as  a  science 
of  application,  and  as  forming  the  true  special  theory  of 
the  geometrical  arts  ?  Considered  in  its  abstract  rela- 
tions, it  could  not  introduce  any  truly  distinct  order  of 
geometrical  speculations.  We  must  not  forget  that,  in 
order  that  a  geometrical  question  should  fall  within  the 
peculiar  domain  of  descriptive  geometry,  it  must  neces- 
sarily have  been  previously  resolved  by  speculative  ge- 
ometry, the  solutions  of  which  then,  as  we  have  seen, 
always  need  to  be  prepared  for  practice  in  such  a  way  as 
to  supply  the  place  of  constructions  in  relief  by  plane 
constructions  ;  a  substitution  which  really  constitutes  the 
only  characteristic  function  of  descriptive  geometry. 

It  is  proper,  however,  to  remark  here,  that,  with  regard 
to  intellectual  education,  the  study  of  descriptive  geome- 
try possesses  an  important  philosophical  peculiarity,  quite 
independent  of  its  high  industrial  utility.  This  is  the 
advantage  which  it  so  pre-eminently  offers — in  habitu- 


DESCRIPTIVE    GEOMETRY.  £23 

ating  the  mind  to  consider  very  complicated  geometrical 
combinations  in  space,  and  to  follow  with  precision  their 
continual  correspondence  with  the  figures  which  are  ac- 
tually traced — of  thus  exercising  to  the  utmost,  in  the 
most  certain  and  precise  manner,  that  important  faculty 
of  the  human  mind  which  is  properly  called  "  imagina- 
tion," and  which  consists,  in  its  elementary  and  positive 
acceptation,  in  representing  to  ourselves,  clearly  and  easi- 
ly, a  vast  and  variable  collection  of  ideal  objects,  as  if 
they  were  really  before  us. 

Finally,  to  complete  the  indication  of  the  general  na- 
ture of  descriptive  geometry  by  determining  its  logical 
character,  we  have  to  observe  that,  while  it  belongs  to 
the  geometry  of  the  ancients  by  the  character  of  its  so- 
lutions, on  the  other  hand  it  approaches  the  geometry  of 
the  moderns  by  the  nature  of  the  questions  which  com- 
pose it.  These  questions  are  in  fact  eminently  remark- 
able for  that  generality  which,  as  we  saw  in  the  prece- 
ding chapter,  constitutes  the  true  fundamental  character 
of  modern  geometry  ;  for  the  methods  used  are  always 
conceived  as  applicable  to  any  figures  whatever,  the  pecu- 
liarity of  each  having  only  a  purely  secondary  influence. 
The  solutions  of  descriptive  geometry  are  then  graphical, 
like  most  of  those  of  the  ancients,  and  at  the  same  time 
general,  like  those  of  the  moderns. 

After  this  important  digression,  we  will  pursue  the 
philosophical  examination  of  special  geometry,  always 
considered  as  reduced  to  its  least  possible  development, 
as  an  indispensable  introduction  to  general  geometry. 
We  have  now  sufficiently  considered  the  graphical  solu- 
tion of  the  fundamental  problem  relating  to  the  right  line 


224  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

— that  is,  the  determination  of  the  different  elements  ot  any 
right-lined  figure  by  means  of  one  another — and  have 
now  to  examine  in  a  special  manner  the  algebraic  solution. 

ALGEBRAIC    SOLUTIONS. 

This  kind  of  solution,  the  evident  superiority  of  which 
need  not  here  be  dwelt  upon,  belongs  necessarily,  by  the 
very  nature  of  the  question,  to  the  system  of  the  ancient 
geometry,  although  the  logical  method  which  is  employed 
causes  it  to  be  generally,  but  very  improperly,  separated 
from  it.  We  have  thus  an  opportunity  of  verifying,  in 
a  very  important  respect,  what  was  established  generally 
in  the  preceding  chapter,  that  it  is  not  by  the  employ- 
ment of  the  calculus  that  the  modern  geometry  is  essen- 
tially to  be  distinguished  from  the  ancient.  The  ancients 
are  in  fact  the  true  inventors  of  the  present  trigonom- 
etry, spherical  as  well  as  rectilinear  ;  it  being  only  much 
less  perfect  in  their  hands,  on  account  of  the  extreme  in- 
feriority of  their  algebraical  knowledge.  It  is,  then,  really 
in  this  chapter,  and  not,  as  it  might  at  first  be  thought, 
in  those  which  we  shall  afterwards  devote  to  the  philo- 
sophical examination  of  general  geometry,  that  it  is  prop- 
er to  consider  the  character  of  this  important  preliminary 
theory,  which  is  usually,  though  improperly,  included  in 
what  is  called  analytical  geometry,  but  which  is  really 
only  a  complement  of  elementary  geometry  properly  so 
called. 

Since  all  right-lined  figures  can  be  decomposed  into 
triangles,  it  is  evidently  sufficient  to  know  how  to  deter- 
mine the  different  elements  of  a  triangle  by  means  of  one 
another,  which  reduces  polygonometry  to  simple  trig- 
onometry. 


TRIGONOMETRY  £25 


TRIGONOMETRY. 

The  difficulty  in  resolving  algebraically  such  a  ques- 
tion as  the  above,  consists  essentially  in  forming,  between 
the  angles  and  the  sides  of  a  triangle,  three  distinct  equa- 
tions ;  which,  when  once  obtained,  will  evidently  reduce  all 
trigonometrical  problems  to  mere  questions  of  analysis. 

How  to  introduce  Angles.  In  considering  the  estab- 
lishment of  these  equations  in  the  most  general  manner, 
we  immediately  meet  with  a  fundamental  distinction 
with  respect  to  the  manner  of  introducing  the  angles 
into  the  calculation,  according  as  they  are  made  to  enter 
directly,  by  themselves  or  by  the  circular  arcs  which  are 
proportional  to  them ;  or  indirectly,  by  the  chords  of 
these  arcs,  which  are  hence  called  their  trigonometrical 
lines.  Of  these  two  systems  of  trigonometry  the  second 
was  of  necessity  the  only  one  originally  adopted,  as  being 
the  only  practicable  one,  since  the  condition  of  geometry 
made  it  easy  enough  to  find  exact  relations  between  the 
sides  of  the  triangles  and  the  trigonometrical  lines  which 
represent  the  angles,  while  it  would  have  been  absolutely 
impossible  at  that  epoch  to  establish  equations  between 
the  sides  and  the  angles  themselves. 

Advantages  of  introducing  Trigonometrical  Lines. 
At  the  present  day,  since  the  solution  can  be  obtained  by 
either  system  indifferently,  that  motive  for  preference  no 
longer  exists  ;  but  geometers  have  none  the  less  persisted 
in  following  from  choice  the  system  primitively  admitted 
from  necessity  ;  for,  the  same  reason  which  enabled  these 
trigonometrical  equations  to  be  obtained  with  much  more 
facility,  must,  in  like  manner,  as  it  is  still  more  easy  to 
conceive  a  priori,  render  these  equations  much  more  sim- 

P 


226  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

pie,  since  they  then  exist  only  between  right  lines,  in- 
stead of  being  established  between  right  lines  and  arcs 
of  circles.  Such  a  consideration  has  so  much  the  more 
importance,  as  the  question  relates  to  formulas  which  are 
eminently  elementary,  and  destined  to  be  continually 
employed  in  all  parts  of  mathematical  science,  as  well 
as  in  all  its  various  applications. 

It  may  be  objected,  however,  that  when  an  angle  is 
given,  it  is,  in  reality,  always  given  by  itself,  and  not  by 
its  trigonometrical  lines  ;  and  that  when  it  is  unknown,  it 
is  its  angular  value  which  is  properly  to  be  determined, 
and  not  that  of  any  of  its  trigonometrical  lines.  It  seems, 
according  to  this,  that  such  lines  are  only  useless  inter- 
mediaries between  the  sides  and  the  angles,  which  have 
to  be  finally  eliminated,  and  the  introduction  of  which 
does  not  appear  capable  of  simplifying  the  proposed  re- 
search. It  is  indeed  important  to  explain,  with  more 
generality  and  precision  than  is  customary,  the  great  real 
utility  of  this  manner  of  proceeding. 

Division  of  Trigonometry  into  two  Parts.  It  con- 
sists in  the  fact  that  the  introduction  of  these  auxiliary 
magnitudes  divides  the  entire  question  of  trigonometry 
into  two  others  essentially  distinct,  one  of  which  has 
for  its  object  to  pass  from  the  angles  to  their  trigono- 
metrical lines,  or  the  converse,  and  the  other  of  which 
proposes  to  determine  the  sides  of  the  triangles  by  the  trig- 
onometrical lines  of  their  angles,  or  the  converse.  Now 
the  first  of  these  two  fundamental  questions  is  evidently 
susceptible,  by  its  nature,  of  being  entirely  treated  and 
reduced  to  numerical  tables  once  for  all,  in  considering 
all  possible  angles,  since  it  depends  only  upon  those  an- 
gles, and  not  at  all  upon  the  particular  triangles  in  which 


TRIGONOMETRY.  227 

they  may  enter  in  each  case ;  while  the  solution  of  the 
second  question  must  necessarily  be  renewed,  at  least  in 
its  arithmetical  relations,  for  each  new  triangle  which  it 
is  necessary  to  resolve.  This  is  the  reason  why  the  first 
portion  of  the  complete  work,  which  would  be  precisely 
the  most  laborious,  is  no  longer  taken  into  the  account, 
being  always  done  in  advance ;  while,  if  such  a  decom- 
position had  not  been  performed,  we  would  evidently  have 
found  ourselves  under  the  obligation  of  recommencing 
the  entire  calculation  in  each  particular  case.  Such  is 
the  essential  property  of  the  present  trigonometrical  sys- 
tem, which  in  fact  would  really  present  no  actual  ad- 
vantage, if  it  was  necessary  to  calculate  continually  the 
trigonometrical  line  of  each  angle  to  be  considered,  or  the 
converse;  the  intermediate  agency  introduced  would  then 
be  more  troublesome  than  convenient. 

In  order  to  clearly  comprehend  the  true  nature  of  this 
conception,  it  will  be  useful  to  compare  it  with  a  still 
more  important  one,  designed  to  produce  an  analogous 
effect  either  in  its  algebraic,  or,  still  more,  in  its  arith- 
metical relations — the  admirable  theory  of  logarithms, 
In  examining  in  a  philosophical  manner  the  influence 
of  this  theory,  we  see  in  fact  that  its  general  result  is 
to  decompose  all  imaginable  arithmetical  operations  into 
two  distinct  parts.  The  first  and  most  complicated  of 
these  is  capable  of  being  executed  in  advance  once  for 
all  (since  it  depends  only  upon  the  numbers  to  be  con- 
sidered, and  not  at  all  upon  the  infinitely  different  com- 
binations into  which  they  can  enter),  and  consists  in  con- 
sidering all  numbers  as  assignable  powers  of  a  constant 
number.  The  second  part  of  the  calculation,  which  must 
of  necessity  be  recommenced  for  each  new  formula  which 


228  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

i 
is  to  have  its  value  determined,  is  thenceforth  reduced 

to  executing  upon  these  exponents  correlative  operations 
which  are  infinitely  more  simple.  I  confine  myself  here 
to  merely  indicating  this  resemblance,  which  any  one  can 
carry  out  for  himself. 

We  must  besides  observe,  as  a  property  (secondary 
at  the  present  day,  but  all-important  at  its  origin)  of  the 
trigonometrical  system  adopted,  the  very  remarkable  cir- 
cumstance that  the  determination  of  angles  by  their  trigo- 
nometrical lines,  or  the  converse,  admits  of  an  arithmetical 
solution  (the  only  one  which  is  directly  indispensable  for 
the  special  destination  of  trigonometry)  without  the  pre- 
vious resolution  of  the  corresponding  algebraic  question. 
It  is  doubtless  to  such  a  peculiarity  that  the  ancients 
owed  the  possibility  of  knowing  trigonometry.  The  in- 
vestigation conceived  in  this  way  was  so  much  the  more 
easy,  inasmuch  as  tables  of  chords  (which  the  ancients 
naturally  took  as  the  trigonometrical  lines)  had  been  pre- 
viously constructed  for  quite  a  different  object,  in  the 
course  of  the  labours  of  Archimedes  on  the  rectification 
of  the  circle,  from  which  resulted  the  actual  determina- 
tion of  a  certain  series  of  chords ;  so  that  when  Hip- 
parchus  subsequently  invented  trigonometry,  he  could 
confine  himself  to  completing  that  operation  by  suitable 
intercalations ;  which  shows  clearly  the  connexion  of  ideas 
in  that  matter. 

The  Increase  of  such  Trigonometrical  Lines.  To 
complete  this  philosophical  sketch  of  trigonometry,  it  is 
proper  now  to  observe  that  the  extension  of  the  same  con- 
siderations which  lead  us  to  replace  angles  or  arcs  of  cir- 
cles by  straight  lines,  with  the  view  of  simplifying  our 
equations,  must  also  lead  us  to  employ  concurrently  sev- 


TRIGONOMETRY.  229 

eral  trigonometrical  lines,  instead  of  confining  ourselves 
to  one  only  (as  did  the  ancients),  so  as  to  perfect  this 
system  by  choosing  that  one  which  will  be  algebraically 
the  most  convenient  on  each  occasion.  In  this  point  of 
view,  it  is  clear  that  the  number  of  these  lines  is  in  itself 
no  ways  limited;  provided  that  they  are  determined  by 
the  arc,  and  that  they  determine  it,  whatever  may  be  the 
law  according  to  which  they  are  derived  from  it,  they  are 
suitable  to  be  substituted  for  it  in  the  equations.  The 
Arabians,  and  subsequently  the  moderns,  in  confining 
themselves  to  the  most  simple  constructions,  have  car- 
ried to  four  or  five  the  number  of  direct  trigonometrical 
lines,  which  might  be  extended  much  farther. 

But  instead  of  recurring  to  geometrical  formations, 
which  would  finally  become  very  complicated,  we  con- 
ceive with  the  utmost  facility  as  many  new  trigono- 
metrical lines  as  the  analytical  transformations  may  re- 
quire, by  means  of  a  remarkable  artifice,  which  is  not 
usually  apprehended  in  a  sufficiently  general  manner. 
It  consists  in  not  directly  multiplying  the  trigonometrical 
lines  appropriate  to  each  arc  considered,  but  in  intro- 
ducing new  ones,  by  considering  this  arc  as  indirectly 
determined  by  all  lines  relating  to  an  arc  which  is  a  very 
simple  function  of  the  first.  It  is  thus,  for  example,  that, 
in  order  to  calculate  an  angle  with  more  facility,  we  will 
determine,  instead  of  its  sine,  the  sine  of  its  half,  or  of 
its  double,  &c.  Such  a  creation  of  indirect  trigono- 
metrical lines  is  evidently  much  more  fruitful  than  all 
the  direct  geometrical  methods  for  obtaining  new  ones. 
We  may  accordingly  say  that  the  number  of  trigono- 
metrical lines  actually  employed  at  the  present  day  by 
geometers  is  in  reality  unlimited,  since  at  every  instant, 


230  ANCIENT  OR  SYNTHETIC  GEOMETRY. 

so  to  say,  the  transformations  of  analysis  may  lead  us  to 
augment  it  by  the  method  which  I  have  just  indicated. 
Special  names,  however,  have  been  given  to  those  only 
of  these  indirect  lines  which  refer  to  the  complement  of 
the  primitive  arc,  the  others  not  occurring  sufficiently 
often  to  render  such  denominations  necessary;  a  cir- 
cumstance which  has  caused  a  common  misconception 
of  the  true  extent  of  the  system  of  trigonometry. 

Study  of  their  Mutual  Relations.  This  multiplicity 
of  trigonometrical  lines  evidently  gives  rise  to  a  third 
fundamental  question  in  trigonometry,  the  study  of  the 
relations  which  exist  between  these  different  lines  ;  since, 
without  such  a  knowledge,  we  could  not  make  use,  for 
our  analytical  necessities,  of  this  variety  of  auxiliary 
magnitudes,  which,  however,  have  no  other  destination. 
It  is  clear,  besides,  from  the  consideration  just  indicated, 
that  this  essential  part  of  trigonometry,  although  simply 
preparatory,  is,  by  its  nature,  susceptible  of  an  indefinite 
extension  when  we  view  it  in  its  entire  generality,  while 
the  two  others  are  circumscribed  within  rigorously  de- 
fined limits. 

It  is  needless  to  add  that  these  three  principal  parts 
of  trigonometry  have  to  be  studied  in  precisely  the  in- 
verse order  from  that  in  which  we  have  seen  them  neces- 
sarily derived  from  the  general  nature  of  the  subject; 
for  the  third  is  evidently  independent  of  the  two  others, 
and  the  second,  of  that  which  was  first  presented — the 
resolution  of  triangles,  properly  so  called — .which  must 
for  that  reason  be  treated  in  the  last  place ;  which  ren- 
dered so  much  the  more  important  the  consideration  of 
their  natural  succession  and  logical  relations  to  one  an- 
other. 


TRIGONOMETRY.  231 

It  is  useless  to  consider  here  separately  spherical  trig- 
onometry, which  cannot  give  rise  to  any  special  philo- 
sophical consideration  ;  since,  essential  as  it  is  by  the  im- 
portance and  the  multiplicity  of  its  us*es,  it  can  be  treated 
at  the  present  day  only  as  a  simple  application  of  rec- 
tilinear trigonometry,  which  furnishes  directly  its  funda- 
mental equations,  by  substituting  for  the  spherical  tri- 
angle the  corresponding  trihedral  angle. 

This  summary  exposition  of  the  philosophy  of  trigo- 
nometry has  been  here  given  in  order  to  render  apparent, 
by  an  important  example,  that  rigorous  dependence  and 
those  successive  ramifications  which  are  presented  by 
what  are  apparently  the  most  simple  questions  of  ele- 
mentary geometry. 

Having  thus  examined  the  peculiar  character  of  spe- 
cial geometry  reduced  to  its  only  dogmatic  destination, 
that  of  furnishing  to  general  geometry  an  indispensable 
preliminary  basis,  we  have  now  to  give  all  our  attention 
to  the  true  science  of  geometry,  considered  as  a  whole, 
in  the  most  rational  manner.  For  that  purpose,  it  is 
necessary  to  carefully  examine  the  great  original  idea  of 
Descartes,  upon  which  it  is  entirely  founded.  This  will 
be  the  object  of  the  following  chapter. 


CHAPTER    III. 

MODERN  OR  ANALYTICAL  GEOMETRY. 

General  (or  Analytical]  geometry  being  entirely 
founded  upon  the  transformation  of  geometrical  consid- 
erations into  equivalent  analytical  considerations,  we 
must  begin  with  examining  directly  and  in  a  thorough 
manner  the  beautiful  conception  by  which  Descartes  has 
established  in  a  uniform  manner  the  constant  possibility 
of  such  a  co-relation.  Besides  its  own  extreme  impor- 
tance as  a  means  of  highly  perfecting  geometrical  science, 
or,  rather,  of  establishing  the  whole  of  it  on  rational 
bases,  the  philosophical  study  of  this  admirable  concep- 
tion must  have  so  much  the  greater  interest  in  our  eyes 
from  its  characterizing  with  perfect  clearness  the  general 
method  to  be  employed  in  organizing  the  relations  of  the 
abstract  to  the  concrete  in  mathematics,  by  the  analyt- 
ical representation  of  natural  phenomena.  There  is  nc 
conception,  in  the  whole  philosophy  of  mathematics 
which  better  deserves  to  fix  all  ou»  attention. 

ANALYTICAL  REPRESENTATION  OF  FIGURES. 

In  order  to  succeed  in  expressing  all  imaginable  geo- 
metrical phenomena  by  simple  analytical  relations,  we 
must  evidently,  in  the  first  place,  establish  a  general 
method  for  representing  analytically  the  subjects  them- 
selves in  which  these  phenomena  are  found,  that  is,  the 
lines  or  the  surfaces  to  be  considered.  The  subject  be- 


REPRESENTATION  OF  FIGURES.    233 

ing  thus  habitually  considered  in  a  purely  analytical 
point  of  view,  we  see  how  it  is  thenceforth  possible  to 
conceive  in  the  same  manner  the  various  accidents  of 
which  it  is  susceptible. 

In  order  to  organize  the  representation  of  geometrical 
figures  by  analytical  equations,  we  must  previously  sur- 
mount a  fundamental  difficulty ;  that  of  reducing  the 
general  elements  of  the  various  conceptions  of  geometry 
to  simply  numerical  ideas ;  in  a  word,  that  of  substitu- 
ting in  geometry  pure  considerations  of  quantity  for  all 
considerations  of  quality. 

Reduction  of  Figure  to  Position.  For  .this  purpose 
let  us  observe,  in  the  first  place,  that  all  geometrical 
ideas  relate  necessarily  to  these  three  universal  catego- 
ries :  the  magnitude,  the  figure,  and  the  position  of  the 
extensions  to  be  considered.  As  to  the  first,  there  is 
evidently  no  difficulty  ;  it  enters  at  once  into  the  ideas 
of  numbers.  With  relation  to  the  second,  it  must  be 
remarked  that  it  will  always  admit  of  being  reduced  to 
the  third.  For  the  figure  of  a  body  evidently  results 
from  the  mutual  position  of  the  different  points  of  which 
it  is  composed,  so  that  the  idea  of  position  necessarily 
comprehends  that  of  figure,  and  every  circumstance  of 
figure  can  be  translated  by  a  circumstance  of  position. 
It  is  in  this  way,  in  fact,  that  the  human  mind  has  pro- 
ceeded in  order  to  arrive  at  the  analytical  representation 
of  geometrical  figures,  their  conception  relating  directly 
only  to  positions.  All  the  elementary  difficulty  is  then 
properly  reduced  to  that  of  referring  ideas  of  situation 
to  ideas  of  magnitude.  Such  is  the  direct  destination 
of  the  preliminary  conception  upon  which  Descartes  has 
established  the  general  system  of  analytical  geometry. 


234    MODERN    OR   ANALYTICAL    GEOMETRY. 

His  philosophical  labour,  in  this  relation,  has  consisted 
simply  in  the  entire  generalization  of  an  elementary  opera- 
tion, which  we  may  regard  as  natural  to  the  human  mind, 
since  it  is  performed  spontaneously,  so  to  say,  in  all 
minds,  even  the  most  uncultivated.  Thus,  when  we 
have  to  indicate  the  situation  of  an  object  without  di- 
rectly pointing  it  out,  the  method  which  we  always  adopt, 
and  evidently  the  only  one  which  can  be  employed,  con- 
sists in  referring  that  object  to  others  which  are  known, 
by  assigning  the  magnitude  of  the  various  geometrical 
elements,  by  which  we  conceive  it  connected  with  the 
known  objects.  These  elements  constitute  what  Des- 
cartes, and  after  him  all  geometers,  have  called  the  co- 
ordinates of  each  point  considered.  They  are  necessarily 
two  in  number,  if  it  is  known  in  advance  in  what  plane 
the  point  is  situated;  and  three,  if  it  may  be  found  in- 
differently in  any  region  of  space.  As  many  different 
constructions  as  can  be  imagined  for  determining  the 
position  of  a  point,  whether  on  a  plane  or  in  space,  so 
many  distinct  systems  of  co-ordinates  may  be  conceived ; 
they  are  consequently  susceptible  of  being  multiplied  to 
infinity.  But,  whatever  may  be  the  system  adopted,  we 
shall  always  have  reduced  the  ideas  of  situation  to  simple 
ideas  of  magnitude,  so  that  we  will  consider  the  change 
in  the  position  of  a  point  as  produced  by  mere  numerical 
variations  in  the  values  of  its  co-ordinates. 

Determination  of  the  Position  of  a  Point.  Consider- 
ing at  first  only  the  least  complicated  case,  that  of  plane 
geometry,  it  is  in  this  way  that  we  usually  determine 
the  position  of  a  point  on  a  plane,  by  its  distances  from 
two  fixed  right  lines  considered  as  known,  which  are 
called  axes,  and  which  are  commonly  supposed  to  be 


REPRESENTATION  OF  FIGURES.    235 

perpendicular  to  each  other.  This  system  is  that  most 
frequently  adopted,  because  of  its  simplicity  ;  but  geom- 
eters employ  occasionally  an  infinity  of  others.  Thus 
the  position  of  a  point  on  a  plane  may  be  determined,  1°, 
by  its  distances  from  two  fixed  points  ;  or,  2°,  by  its  dis- 
tance from  a  single  fixed  point,  and  the  direction  of  that 
distance,  estimated  by  the  greater  or  less  angle  which  it 
makes  with  a  fixed  right  line,  which  constitutes  the  sys- 
tem of  what  are  called  polar  co-ordinates,  the  most  fre- 
quently used  after  the  system  first  mentioned;  or,  3°,  by 
the  angles  which  the  right  lines  drawn  from  the  variable 
point  to  two  fixed  points  make  with  the  right  line  which 
joins  these  last;  or,  4°,  by  the  distances  from  that  point 
to  a  fixed  right  line  and  a  fixed  point,  &c.  In  a  word, 
there  is  no  geometrical  figure  whatever  from  which  it  is 
not  possible  to  deduce  a  certain  system  of  co-ordinates 
more  or  less  susceptible  of  being  employed. 

A  general  observation,  which  it  is  important  to  make 
in  this  connexion,  is,  that  every  system  of  co-ordinates  is 
equivalent  to  determining  a  point,  in  plane  geometry,  by 
the  intersection  of  two  lines,  each  of  which  is  subjected 
to  certain  fixed  conditions  of  determination ;  a  single 
one  of  these  conditions  remaining  variable,  sometimes 
the  one,  sometimes  the  other,  according  to  the  system 
considered.  We  could  not,  indeed,  conceive  any  other 
means  of  constructing  a  point  than  to  mark  it  by  the 
meeting  of  two  lines.  Thus,  in  the  most  common  sys- 
tem, that  of  rectilinear  co-ordinates,  properly  so  called, 
the  point  is  determined  by  the  intersection  of  two  right 
lines,  each  of  which  remains  constantly  parallel  to  a 
fixed  axis,  at  a  greater  or  less  distance  from  it ;  in  the 
polar  system,  the  position  of  the  point  is  marked  by  the 


236  MODERN  OR  ANALYTICAL  GEOMETRY. 

meeting  of  a  circle,  of  variable  radius  and  fixed  centre, 
with  a  movable  right  line  compelled  to  turn  about  this 
centre  :  in  other  systems,  the  required  point  might  be 
designated  by  the  intersection  of  two  circles,  or  of  any 
other  two  lines,  &c.  In  a  word,  to  assign  the  value  of 
one  of  the  co-ordinates  of  a  point  in  any  system  what- 
ever, is  always  necessarily  equivalent  to  determining  a 
certain  line  on  which  that  point  must  be  situated.  The 
geometers  of  antiquity  had  already  made  this  essential 
remark,  which  served  as  the  base  of  their  method  of 
geometrical  loci,  of  which  they  made  so  happy  a  use  to 
direct  their  researches  in  the  resolution  of  determinate 
problems,  in  considering  separately  the  influence  of  each 
of  the  two  conditions  by  which  was  defined  each  point 
constituting  the  object,  direct  or  indirect,  of  the  proposed 
question.  It  was  the  general  systematization  of  this 
method  which  was  the  immediate  motive  of  the  labours 
of  Descartes,  which  led  him  to  create  analytical  geom- 
etry. 

After  having  clearly  established  this  preliminary  con- 
ception— by  means  of  which  ideas  of  position,  and  thence, 
implicitly,  all  elementary  geometrical  conceptions  are  ca- 
pable of  being  reduced  to  simple  numerical  considera- 
tions— it  is  easy  to  form  a  direct  conception,  in  its  entire 
generality,  of  the  great  original  idea  of  Descartes,  rela- 
tive to  the  analytical  representation  of  geometrical  fig- 
ures :  it  is  this  which  forms  the  special  object  of  this 
chapter.  I  will  continue  to  consider  at  first,  for  more 
facility,  only  geometry  of  two  dimensions,  which  alone 
was  treated  by  Descartes ;  and  will  afterwards  examine 
separately,  under  the  same  point  of  view,  the  theory  of 
surfaces  and  curves  of  double  curvature. 


PLANE    CURVES.  237 


PLANE    CURVES. 

Expression  of  Lines  by  Equations.  In  accordance 
with  the  manner  of  expressing  analytically  the  position 
of  a  point  on  a  plane,  it  can  be  easily  established  that, 
by  whatever  property  any  line  may  be  denned,  that  defi- 
nition always  admits  of  being  replaced  by  a  correspond- 
ing equation  between  the  two  variable  co-ordinates  of  the 
point  which  describes  this  line ;  an  equation  which  will 
be  thenceforth  the  analytical  representation  of  the  pro- 
posed line,  every  phenomenon  of  which  will  be  translated 
by  a  certain  algebraic  modification  of  its  equation.  Thus, 
if  we  suppose  that  a  point  moves  on  a  plane  without  its 
course  being  in  any  manner  determined,  we  shall  evi- 
dently have  to  regard  its  co-ordinates,  to  whatever  system 
they  may  belong,  as  two  variables  entirely  independent 
of  one  another.  But  if,  on  the  contrary,  this  point  is 
compelled  to  describe  a  certain  line,  we  shall  necessarily 
be  compelled  to  conceive  that  its  co-ordinates,  in  all  the 
positions  which  it  can  take,  retain  a  certain  permanent 
and  precise  relation  to  each  other,  which  is  consequently 
susceptible  of  being  expressed  by  a  suitable  equation ; 
which  will  become  the  very  clear  and  very  rigorous  an- 
alytical definition  of  the  line  under  consideration,  since 
it  will  express  an  algebraical  property  belonging  exclu- 
sively to  the  co-ordinates  of  all  the  points  of  this  line. 
It  is  clear,  indeed,  that  when  a  point  is  not  subjected  to 
any  condition,  its  situation  is  not  determined  except  in 
giving  at  once  its  two  co-ordinates,  independently  of  each 
other  ;  while,  when  the  point  must  continue  upon  a  de- 
fined line,  a  single  co-ordinate  is  sufficient  for  complete- 
ly fixing  its  position.  The  second  co-ordinate  is  then  a 


238  MODERN  OR  ANALYTICAL  GEOMETRY. 

determinate  function  of  the  first ;  or,  in  other  words, 
there  must  exist  between  them  a  certain  equation,  of  a 
nature  corresponding  to  that  of  the  line  on  which  the 
point  is  compelled  to  remain.  In  a  word,  each  of  the 
co-ordinates  of  a  point  requiring  it  to  be  situated  on  a 
certain  line,  we  conceive  reciprocally  that  the  condition, 
on  the  part  of  a  point,  of  having  to  belong  to  a  line  de- 
fined in  any  manner  whatever,  is  equivalent  to  assigning 
the  value  of  one  of  the  two  co-ordinates ;  which  is  found 
in  that  case  to  be  entirely  dependent  on  the  other.  The 
analytical  relation  which  expresses  this  dependence  may 
be  more  or  less  difficult  to  discover,  but  it  must  evi- 
dently be  always  conceived  to  exist,  even  in  the  cases  in 
which  our  present  means  may  be  insufficient  to  make  it 
known.  It  is  by  this  simple  consideration  that  we  may 
demonstrate,  in  an  entirely  general  manner — independ- 
ently of  the  particular  verifications  on  which  this  funda- 
mental conception  is  ordinarily  established  for  each  spe- 
cial definition  of  a  line — the  necessity  of  the  analytical 
representation  of  lines  by  equations. 

Expression  of  Equations  by  Lines.  Taking  up  again 
the  same  reflections  in  the  inverse  direction,  we  could 
show  as  easily  the  geometrical  necessity  of  the  represent- 
ation of  every  equation  of  two  variables,  in  a  determinate 
system  of  co-ordinates,  by  a  certain  line  ;  of  which  such 
a  relation  would  be,  in  the  absence  of  any  other  known 
property,  a  very  characteristic  definition,  the  scientific 
destination  of  which  will  be  to  fix  the  attention  directly 
upon  the  general  course  of  the  solutions  of  the  equation, 
which  will  thus  be  noted  in  the  most  striking  and  the 
most  simple  manner.  This  picturing  of  equations  is  one 
of  the  most  important  fundamental  advantages  of  ana- 


PLANE    CURVES.  239 

lytical  geometry,  which  has  thereby  reacted  in  the  high- 
est degree  upon  the  general  perfecting  of  analysis  itself ; 
not  only  by  assigning  to  purely  abstract  researches  a 
clearly  determined  object  and  an  inexhaustible  career, 
but,  in  a  still  more  direct  relation,  by  furnishing  a  new 
philosophical  medium  for  analytical  meditation  which 
could  not  be  replaced  by  any  other.  In  fact,  the  purely 
algebraic  discussion  of  an  equation  undoubtedly  makes 
known  its  solutions  in  the  most  precise  manner,  but  in 
considering  them  only  one  by  one,  so  that  in  this  way 
no  general  view  of  them  could  be  obtained,  except  as  the 
final  result  of  a  long  and  laborious  series  of  numerical 
comparisons.  On  the  other  hand,  the  geometrical  locus 
of  the  equation,  being  only  designed  to  represent  distinct- 
ly and  with  perfect  clearness  the  summing  up  of  all  these 
comparisons,  permits  it  to  be  directly  considered,  without 
paying  any  attention  to  the  details  which  have  furnished 
it.  It  can  thereby  suggest  to  our  mind  general  analyt- 
ical views,  which  we  should  have  arrived  at  with  much 
difficulty  in  any  other  manner,  for  want  of  a  means  of 
clearly  characterizing  their  object.  It  is  evident,  for  ex- 
ample, that  the  simple  inspection  of  the  logarithmic 
curve,  or  of  the  curve  y  =  sin.  x,  makes  us  perceive 
much  more  distinctly  the  general  manner  of  the  varia- 
tions of  logarithms  with  respect  to  their  numbers,  or  of 
sines  with  respect  to  their  arcs,  than  could  the  most  at- 
tentive study  of  a  table  of  logarithms  or  of  natural  sines. 
It  is  well  known  that  this  method  has  become  entirely 
elementary  at  the  present  day,  and  that  it  is  employed 
whenever  it  is  desired  to  get  a  clear  idea  of  the  general 
character  of  the  law  which  reigns  in  a  series  of  precise 
observations  of  any  kind  whatever. 


240'MODERN    OR    ANALYTICAL    GEOMETRY. 

Any  Change  in  the  Line  causes  a  Change  in  the 
Equation.  Returning  to  the  representation  of  lines  by 
equations,  which  is  our  principal  object,  we  see  that  this 
representation  is,  by  its  nature,  so  faithful,  that  the  line 
could  not  experience  any  modification,  however  slight  it 
might  be,  without  causing  a  corresponding  change  in  the 
equation.  This  perfect  exactitude  even  gives  rise  often- 
times to  special  difficulties;  for  since,  in  our  system  of 
analytical  geometry,  the  mere  displacements  of  lines  af- 
fect the  equations,  as  well  as  their  real  variations  in  mag- 
nitude or  form,  we  should  be  liable  to  confound  them 
with  one  another  in  our  analytical  expressions,  if  geom- 
eters had  not  discovered  an  ingenious  method  designed 
expressly  to  always  distinguish  them.  This  method  is 
founded  on  this  principle,  that  although  it  is  impossible 
to  change  analytically  at  will  the  position  of  a  line  with 
respect  to  the  axes  of  the  co-ordinates,  we  can  change  in 
any  manner  whatever  the  situation  of  the  axes  them- 
selves, which  evidently  amounts  to  the  same ;  then,  by 
the  aid  of  the  very  simple  general  formula  by  which  this 
transformation  of  the  axes  is  produced,  it  becomes  easy 
to  discover  whether  two  different  equations  are  the  ana- 
lytical expressions  of  only  the  same  line  differently  situ- 
ated, or  refer  to  truly  distinct  geometrical  loci ;  since,  in 
the  former  case,  one  of  them  will  pass  into  the  other  by 
suitably  changing  the  axes  or  the  other  constants  of  the 
system  of  co-ordinates  employed.  It  must,  moreover,  be 
remarked  on  this  subject,  that  general  inconveniences  of 
this  nature  seem  to  be  absolutely  inevitable  in  analytical 
geometry  ;  for,  since  the  ideas  of  position  are,  as  we  have 
seen,  the  only  geometrical  ideas  immediately  reducible  to 
numerical  considerations,  and  the  conceptions  of  figure 


PLANE   CURVES.  241 

cannot  be  thus  reduced,  except  by  seeing  in  them  rela- 
tions of  situation,  it  is  impossible  for  analysis  to  escape 
confounding,  at  first,  the  phenomena  of  figure  with  sim- 
ple phenomena  of  position,  which  alone  are  directly  ex- 
pressed by  the  equations. 

Every  Definition  of  a  Line  is  an  Equation.  In  or- 
der to  complete  the  philosophical  explanation  of  the  fun- 
damental conception  which  serves  as  the  base  of  analyt- 
ical geometry,  I  think  that  I  should  here  indicate  a  new 
general  consideration,  which  seems  to  me  particularly 
well  adapted  for  putting  in  the  clearest  point  of  view  this 
necessary  representation  of  lines  by  equations  with  two 
variables.  It  consists  in  this,  that  not  only,  as  we  have 
shown,  must  every  defined  line  necessarily  give  rise  to  a 
certain  equation  between  the  two  co-ordinates  of  any  one 
of  its  points,  but,  still  farther,  every  definition  of  a  line 
may  be  regarded  as  being  already  of  itself  an  equation  of 
that  line  in  a  suitable  system  of  co-ordinates. 

It  is  easy  to  establish  this  principle,  first  making  a 
preliminary  logical  distinction  with  respect  to  different 
kinds  of  definitions.  The  rigorously  indispensable  con- 
dition of  every  definition  is  that  of  distinguishing  the  ob- 
ject defined  from  all  others,  by  assigning  to  it  a  property 
which  belongs  to  it  exclusively.  But  this  end  may  be 
generally  attained  in  two  very  different  ways ;  either  by 
a  definition  which  is  simply  characteristic,  that  is,  in- 
dicative of  a  property  which,  although  truly  exclusive, 
does  not  make  known  the  mode  of  generation  of  the  nh- 
ject ;  or  by  a  definition  which  is  really  explanatory,  that 
is,  which  characterizes  the  object  by  a  property  which  ex- 
presses one  of  its  modes  of  generation.  For  example,  in 
considering  the  circle  as  the  line,  which,  under  the  same 

Q 


£42  MODERN  OR  ANALYTICAL  GEOMETRY. 

contour,  contains  the  greatest  area,  we  have  evidently  a 
definition  of  the  first  kind  ;  while  in  choosing  the  prop- 
erty of  its  having  all  its  points  eqally  distant  from  a  fixed 
point,  we  have  a  definition  of  the  second  kind.  It  is,  be- 
sides, evident,  as  a  general  principle,  that  even  when  any 
object  whatever  is  known  at  first  only  by  a  characteristic 
definition,  we  ought,  nevertheless,  to  regard  it  as  suscep- 
tible of  explanatory  definitions,  which  the  farther  study 
of  the  object  would  necessarily  lead  us  to  discover. 

This  being  premised,  it  is  clear  that  the  general  ob- 
servation above  made,  which  represents  every  definition 
of  a  line  as  being  necessarily  an  equation  of  that  line  in 
a  certain  system  of  co-ordinates,  cannot  apply  to  defini- 
tions which  are  simply  characteristic ;  it  is  to  be  un- 
derstood only  of  definitions  which  are  truly  explanatory. 
But,  in  considering  only  this  class,  the  principle  is  easy 
to  prove.  In  fact,  it  is  evidently  impossible  to  define  the 
generation  of  a  line  without  specifying  a  certain  relation 
between  the  two  simple  motions  of  translation  or  of  rota- 
tion, into  which  the  motion  of  the  point  which  describes  it 
will  be  decomposed  at  each  instant.  Now  if  we  form  the 
most  general  conception  of  what  constitutes  a  system  of 
co-ordinates,  and  admit  all  possible  systems,  it  is  clear 
that  such  a  relation  will  be  nothing  else  but  the  equation 
of  the  proposed  line,  in  a  system  of  co-ordinates  of  a  na- 
ture corresponding  to  that  of  the  mode  of  generation  con- 
sidered. Thus,  for  example,  the  common  definition  of 
the  circle  may  evidently  be  regarded  as  being  immedi- 
ately the  polar  equation  of  this  curve,  taking  the  centre 
of  the  circle  for  the  pole.  In  the  same  way,  the  ele- 
mentary definition  of  the  ellipse  or  of  the  hyperbola — as 
being  the  curve  generated  by  a  point  which  moves  in 


PLANE    CURVES.  £43 

such  a  manner  that  the  sum  or  the  difference  of  its  dis- 
tances from  two  fixed  points  remains  constant — gives  at 
once,  for  either  the  one  or  the  other  curve,  the  equation 
y+x=c,  taking  for  the  system  of  co-ordinates  that  in 
which  the  position  of  a  point  would  be  determined  by  its 
distances  from  two  fixed  points,  and  choosing  for  these 
poles  the  two  given  foci.  In  like  manner,  the  common 
definition  of,  any  cycloid  would  furnish  directly,  for  that 
curve,  the  equation  y=mx  ;  adopting  as  the  co-ordinates 
of  each  point  the  arc  which  it  marks  upon  a  circle  of  inva- 
riable radius,  measuring  from  the  point  of  contact  of  that 
circle  with  a  fixed  line,  and  the  rectilinear  distance  from 
that  point  of  contact  to  a  certain  origin  taken  on  that 
right  line.  We  can  make  analogous  and  equally  easy  ver- 
ifications with  respect  to  the  customary  definitions  of  spi- 
rals, of  epicycloids,  &c.  We  shall  constantly  find  that 
there  exists  a  certain  system  of  co-ordinates,  in  which  we 
immediately  obtain  a  very  simple  equation  of  the  pro- 
posed line,  by  merely  writing  algebraically  the  condition 
imposed  by  the  mode  of  generation  considered. 

Besides  its  direct  importance  as  a  means  of  rendering 
perfectly  apparent  the  necessary  representation  of  every 
line  by  an  equation,  the  preceding  consideration  seems  to 
me  to  possess  a  true  scientific  utility,  in  characterizing 
with  precision  the  principal  general  difficulty  which  oc- 
curs in  the  actual  establishment  of  these  equations,  and  in 
consequently  furnishing  an  interesting  indication  with  re- 
spect to  the  course  to  be  pursued  in  inquiries  of  this  kind, 
which,  by  their  nature,  could  not  admit  of  complete  and 
invariable  rules.  In  fact,  since  any  definition  whatever 
of  a  line,  at  least  among  those  which  indicate  a  mode  of 
generation,  furnishes  directly  the  equation  of  that  line  in 


244  MODERN  OR  ANALYTICAL  GEOMETRY. 

a  certain  system  of  co-ordinates,  or,  rather,  of  itself  con- 
stitutes that  equation,  it  follows  that  the  difficulty  which 
we  often  experience  in  discovering  the  equation  of  a 
curve,  by  means  of  certain  of  its  characteristic  properties, 
a  difficulty  which  is  sometimes  very  great,  must  proceed 
essentially  only  from  the  commonly  imposed  condition  of 
expressing  this  curve  analytically  by  the  aid  of  a  desig- 
nated system  of  co-ordinates,  instead  of  admitting  indif- 
ferently all  possible  systems.  These  different  systems 
cannot  be  regarded  in  analytical  geometry  as  being  all 
equally  suitable ;  for  various  reasons,  the  most  impor- 
tant of  which  will  be  hereafter  discussed,  geometers  think 
that  curves  should  almost  always  be  referred,  as  far  as  is 
possible,  to  rectilinear  co-ordinates,  properly  so  called. 
Now  we  see,  from  what  precedes,  that  in  many  cases  these 
particular  co-ordinates  will  not  be  those  with  reference  to 
which  the  equation  of  the  curve  will  be  found  to  be  di- 
rectly established  by  the  proposed  definition.  The  prin- 
cipal difficulty  presented  by  the  formation  of  the  equation 
of  a  line  really  consists,  then,  in  general,  in  a  certain 
transformation  of  co-ordinates.  It  is  undoubtedly  true 
that  this  consideration  does  not  subject  the  establishment 
of  these  equations  to  a  truly  complete  general  method,  the 
success  of  which  is  always  certain  ;  which,  from  the  very 
nature  of  the  subject,  is  evidently  chimerical :  but  such  a 
view  may  throw  much  useful  light  upon  the  course  which 
it  is  proper  to  adopt,  in  order  to  arrive  at  the  end  pro- 
posed. Thus,  after  having  in  the  first  place  formed  the 
preparatory  equation,  which  is  spontaneously  derived 
from  the  definition  which  we  are  considering,  it  will  be 
necessary,  in  order  to  obtain  the  equation  belonging  to 
the  system  of  co-ordinates  which  must  be  finally  admit- 


CHOICE   OF   CO-ORDINATES.  245 

ted,  to  endeavour  to  express  in  a  function  of  these  last  co- 
ordinates those  which  naturally  correspond  to  the  given 
mode  of  generation.  It  is  upon  this  last  labour  that  it 
is  evidently  impossible  to  give  invariable  and  precise  pre- 
cepts. We  can  only  say  that  we  shall  have  so  many 
more  resources  in  this  matter  as  we  shall  know  more  of 
true  analytical  geometry,  that  is,  as  we  shall  know  the 
algebraical  expression  of  a  greater  number  of  different  al- 
gebraical phenomena. 

CHOICE    OF    CO-ORDINATES. 

In  order  to  complete  the  philosophical  exposition  of  the 
conception  which  serves  as  the  base  of  analytical  geom- 
etry, I  have  yet  to  notice  the  considerations  relating  to 
the  choice  of  the  system  of  co-ordinates  which  is  in  gen- 
eral the  most  suitable.  They  will  give  .the  rational  ex- 
planation of  the  preference  unanimously  accorded  to  the 
ordinary  rectilinear  system  ;  a  preference  which  has  hith- 
erto been  rather  the  effect  of  an  empirical  sentiment  of 
the  superiority  of  this  system,  than  the  exact  result  of  a 
direct  and  thorough  analysis. 

Two  different  Points  of  View.  In  order  to  decide 
clearly  between  all  the  different  systems  of  co-ordinates, 
it  is  indispensable  to  distinguish  with  care  the  two  gen- 
eral points  of  view,  the  converse  of  one  another,  which 
belong  to  analytical  geometry ;  namely,  the  relation  of 
algebra  to  geometry,  founded  upon  the  representation  of 
lines  by  equations  ;  and,  reciprocally,  the  relation  of  ge- 
ometry to  algebra,  founded  on  the  representation  of  equa-^ 
tions  by  lines. 

It  is  evident  that  in  every  investigation  of  general  ge- 
ometry these  two  fundamental  points  of  view  are  of  ne- 


246  MODERN  OR  ANALYTICAL  GEOMETRY. 

oessity  always  found  combined,  since  we  have  always  to 
pass  alternately,  and  at  insensible  intervals,  so  to  say, 
from  geometrical  to  analytical  considerations,  and  from 
analytical  to  geometrical  considerations.  But  the  ne- 
cessity of  here  temporarily  separating  them  is  none  the 
less  real ;  for  the  answer  to  the  question  of  method  which 
we  are  examining  is,  in  fact,  as  we  shall  see  presently, 
very  far  from  being  the  same  in  both  these  relations,  so 
that  without  this  distinction  we  could  not  form  any  clear 
idea  of  it. 

1.  Representation  of  Lines  by  Equations.     Under  the 
first  point  of  view — the  representation  of  lines  by  equa- 
tions— the  only  reason  which  could  lead  us  to  prefer  one 
system  of  co-ordinates  to  another  would  be  the  greater 
simplicity  of  the  equation  of  each  line,  and  greater  facil- 
ity in  arriving  at  it.     Now  it  is  easy  to  see  that  there  does 
not  exist,  and  could  not  be  expected  to  exist,  any  system 
of  co-ordinates  deserving  in  that  respect  a  constant  pref- 
erence over  all  others.      In  fact,  we  have  above  remarked 
that  for  each  geometrical  definition  proposed  we  can  con- 
ceive a  system  of  co-ordinates  in  which  the  equation  of 
the  line  is  obtained  at  once,  and  is  necessarily  found  to 
be  also  very  simple ;  and  this  system,  moreover,  inevita- 
bly varies  with  the  nature  of  the  characteristic  property 
under  consideration.      The  rectilinear  system  could  not, 
therefore,  be  constantly  the  most  advantageous  for  this  ob- 
ject, although  it  may  often  be  very  favourable  ;  there  is 
probably  no  system  which,  in  certain  particular  cases, 
should  not  be  preferred  to  it,  as  well  as  to  every  other. 

2.  Representation  of  Equations  by  Lines.    It  is  by  no 
means  so,  however,  under  the  second  point  of  view.     We 
can,  indeed,  easily  establish,  as  a  general  principle,  that 


CHOICE   OF  CO-ORDINATES.  £47 

the  ordinary  rectilinear  system  must  necessarily  be  bet- 
ter adapted  than  any  other  to  the  representation  of  equa- 
tions by  the  corresponding  geometrical  loci ;  that  is  to 
say,  that  this  representation  is  constantly  more  simple 
and  more  faithful  in  it  than  in  any  other. 

Let  us  consider,  for  this  object,  that,  since  every  sys- 
tem of  co-ordinates  consists  in  determining  a  point  by  the 
intersection  of  two  lines,  the  system  adapted  to  furnish 
the  most  suitable  geometrical  loci  must  be  that  in  which 
these  two  lines  are  the  simplest  possible  ;  a  consideration 
which  confines  our  choice  to  the  rectilinear  system.  In 
truth,  there  is  evidently  an  infinite  number  of  systems 
which  deserve  that  name,  that  is  to  say,  which  employ 
only  right  lines  to  determine  points,  besides  the  ordinary 
system  which  assigns  the  distances  from  two  fixed  lines 
as  co-ordinates;  such,  for  example,  would  be  that  in 
which  the  co-ordinates  of  each  point  should  be  the  two 
angles  which  the  right  lines,  which  go  from  that  point  to 
two  fixed  points,  make  with  the  right  line,  which  joins 
these  last  points :  so  that  this  first  consideration  is  not 
rigorously  sufficient  to  explain  the  preference  unanimous- 
ly given  to  the  common  system.  But  in  examining  in  a 
more  thorough  manner  the  nature  of  every  system  of  co- 
ordinates, we  also  perceive  that  each  of  the  two  lines, 
whose  meeting  determines  the  point  considered,  must 
necessarily  offer  at  every  instant,  among  its  different  con- 
ditions of  determination,  a  single  variable  condition,  which 
gives  rise  to  the  corresponding  co-ordinate,  all  the  rest 
being  fixed,  and  constituting  the  axes  of  the  system, 
taking  this  term  in  its  most  extended  mathematical  ac- 
ceptation. The  variation  is  indispensable,  in  order  that 
we  may  be  able  to  consider  all  possible  positions ;  and 


248  MODERN  OR  ANALYTICAL  GEOMETRY. 

the  fixity  is  no  less  so,  in  order  that  there  may  exist 
means  of  comparison.  Thus,  in  all  rectilinear  systems, 
each  of  the  two  right  lines  will  be  subjected  to  a  fixed 
condition,  and  the  ordinate  will  result  from  the  variable 
condition. 

Superiority  of  rectilinear  Co-ordinates.  From  these 
considerations  it  is  evident,  as  a  general  principle,  that 
the  most  favourable  system  for  the  construction  of  geo- 
metrical loci  will  necessarily  be  that  in  which  the  vari- 
able condition  of  each  right  line  shall  be  the  simplest 
possible ;  the  fixed  condition  being  left  free  to  be  made 
complex,  if  necessary  to  attain  that  object.  Now,  of 
all  possible  manners  of  determining  two  movable  right 
lines,  the  easiest  to  follow  geometrically  is  certainly  that 
in  which,  the  direction  of  each  right  line  remaining  in- 
variable, it  only  approaches  or  recedes,  more  or  less,  to 
or  from  a  constant  axis.  It  would  be,  for  example,  evi- 
dently more  difficult  to  figure  to  one's  self  clearly  the 
changes  of  place  of  a  point  which  is  determined  by  the 
intersection  of  two  right  lines,  which  each  turn  around 
a  fixed  point,  making  a  greater  or  smaller  angle  with  a 
certain  axis,  as  in  the  system  of  co-ordinates  previously 
noticed.  Such  is  the  true  general  explanation  of  the 
fundamental  property  possessed  by  the  common  rectilin- 
ear system,  of  being  better  adapted  than  any  other  to  the 
geometrical  representation  of  equations,  inasmuch  as  it 
is  that  one  in  which  it  is  the  easiest  to  conceive  the 
change  of  place  of  a  point  resulting  from  the  change  in 
the  value  of  its  co-ordinates.  In  order  to  feel  clearly  all 
the  force  of  this  consideration,  it  would  be  sufficient  to 
carefully  compare  this  system  with  the  polar  system,  in 
which  this  geometrical  image,  so  simple  and  so  easy  to 


CHOICE   OF   CO-ORDINATES.  £49 

follow,  of  two  right  lines  moving  parallel,  each  one  of 
them,  to  its  corresponding  axis,  is  replaced  by  the  com- 
plicated picture  of  an  infinite  series  of  concentric  cir- 
cles, cut  by  a  right  line  compelled  to  turn  about  a  fixed 
point.  It  is,  moreover,  easy  to  conceive  in  advance  what 
must  be  the  extreme  importance  to  analytical  geometry 
of  a  property  so  profoundly  elementary,  which,  for  that 
reason,  must  be  recurring  at  every  instant,  and  take  a 
progressively  increasing  value  in  all  labours  of  this  kind. 
Perpendicularity  of  the  Axes.  In  pursuing  farther 
the  consideration  which  demonstrates  the  superiority  of 
the  ordinary  system  of  co-ordinates  over  any  other  as  to 
the  representation  of  equations,  we  may  also  take  notice 
of  the  utility  for  this  object  of  the  common  usage  of  tak- 
ing the  two  axes  perpendicular  to  each  other,  whenever 
possible,  rather  than  with  any  other  inclination.  As  re- 
gards the  representation  of  lines  by  equations,  this  sec- 
ondary circumstance  is  no  more  universally  proper  than 
we  have  seen  the  general  nature  of  the  system  to  be ; 
since,  according  to  the  particular  occasion,  any  other  in- 
clination of  tfie  axes  may  deserve  our  preference  in  that 
respect.  But,  in  the  inverse  point  of  view,  it  is  easy  to 
see  that  rectangular  axes  constantly  permit  us  to  repre- 
sent equations  in  a  more  simple  and  even  more  faithful 
manner ;  for,  with  oblique  axes,  space  being  divided  by 
them  into  regions  which  no  longer  have  a  perfect  identity, 
it  follows  that,  if  the  geometrical  locus  of  the  equation 
extends  into  all  these  regions  at  once,  there  will  be  pre- 
sented, by  reason  merely  of  this  inequality  of  the  angles, 
differences  of  figure  which  do  not  correspond  to  any 
analytical  diversity,  and  will  necessarily  alter  the  rigor- 
ous exactness  of  the  representation,  by  being  confounded 


250  MODERN  OR  ANALYTICAL  GEOMETRY. 

with  the  proper  results  of  the  algebraic  comparisons. 
For  example,  an  equation  like  xm+ym= c,  which,  by  its 
perfect  symmetry,  should  evidently  give  a  curve  com- 
posed of  four  identical  quarters,  will  be  represented,  on 
the  contrary,  if  we  take  axes  not  rectangular,  by  a  geo- 
metric locus,  the  four  parts  of  which  will  be  unequal. 
It  is  plain  that  the  only  means  of  avoiding  all  incon- 
veniences of  this  kind  is  to  suppose  the  angle  of  the  two 
axes  to  be  a  right  angle. 

The  preceding  discussion  clearly  shows  that,  although 
the  ordinary  system  of  rectilinear  co-ordinates  has  no  con- 
stant superiority  over  all  others  in  one  of  the  two  funda- 
mental points  of  view  which  are  continually  combined  in 
analytical  geometry,  yet  as,  on  the  other  hand,  it  is  not 
constantly  inferior,  its  necessary  and  absolute  greater 
aptitude  for  the  representation  of  equations  must  cause 
it  to  generally  receive  the  preference ;  although  it  may 
evidently  happen,  in  some  particular  cases,  that  the  ne- 
cessity of  simplifying  equations  and  of  obtaining  them 
more  easily  may  determine  geometers  to  adopt  a  less 
perfect  system.  The  rectilinear  system  is,  therefore,  the 
one  by  means  of  which  are  ordinarily  constructed  the 
most  essential  theories  of  general  geometry,  intended  to 
express  analytically  the  most  important  geometrical  phe- 
nomena. When  it  is  thought  necessary  to  choose  some 
other,  the  polar  system  is  almost  always  the  one  which 
is  fixed  upon,  this  system  being  of  a  nature  sufficiently 
opposite  to  that  of  the  rectilinear  system  to  cause  the 
equations,  which  are  too  complicated  with  respect  to  the 
latter,  to  become,  in  general,  sufficiently  simple  with  re- 
spect to  the  other.  Polar  co-ordiuates,  moreover,  have 
often  the  advantage  of  admitting  of  a  more  direct  and 


SURFACES.  251 

natural  concrete  signification;  as  is  the  case  in  mechan- 
ics, for  the  geometrical  questions  to  which  the  theory  of 
circular  movement  gives  rise,  and  in  almost  all  the  cases 
of  celestial  geometry. 

In  order  to  simplify  the  exposition,  we  have  thus  far 
considered  the  fundamental  conception  of  analytical  ge- 
ometry only  with  respect  to  plane  curves,  the  general 
study  of  which  was  the  only  object  of  the  great  philo- 
sophical renovation  produced  by  Descartes.  To  com- 
plete this  important  explanation,  we  have  now  to  show 
summarily  how  this  elementary  idea  was  extended  by 
Clairaut,  about  a  century  afterwards,  to  the  general 
study  of  surfaces  and  curves  of  double  curvature.  The 
considerations  which  have  been  already  given  will  per- 
mit me  to  limit  myself  on  this  subject  to  the  rapid  ex- 
amination of  what  is  strictly  peculiar  to  this  new  case. 

SURFACES. 

Determination  of  a  Point  in  Space.  The  complete 
analytical  determination  of  a  point  in  space  evidently  re- 
quires the  values  of  three  co-ordinates  to  be  assigned ;  as, 
for  example,  in  the  system  which  is  generally  adopted, 
and  which  corresponds  to  the  rectilinear  system  of  plane 
geometry,  distances  from  the  point  to  three  fixed  planes, 
usually  perpendicular  to  one  another  ;  which  presents  the 
point  as  the  intersection  of  three  planes  whose  direction 
is  invariable.  We  might  also  employ  the  distances  from 
the  movable  point  to  three  fixed  points,  which  would 
determine  it  by  the  intersection  of  three  spheres  with  a 
common  centre.  In  like  manner,  the  position  of  a  point 
would  be  defined  by  giving  its  distance  from  a  fixed  point, 


252  MODERN  OR  ANALYTICAL  GEOMETRY. 

and  the  direction  of  that  distance,  by  means  of  the  twc 
angles  which  this  right  line  makes  with  two  invariable 
axes ;  this  is  the  polar  system  of  geometry  of  three  di- 
mensions ;  the  point  is  then  constructed  by  the  inter- 
section of  a  sphere  having  a  fixed  centre,  with  two  right 
cones  with  circular  bases,  whose  axes  and  common  sum- 
mit do  not  change.  In  a  word,  there  is  evidently,  in  this 
case  a.t  least,  the  same  infinite  variety  among  the  vari- 
ous possible  systems  of  co-ordinates  which  we  have  al- 
ready observed  in  geometry  of  two  dimensions.  In  gen- 
eral, we  have  to  conceive  a  point  as  being  always  deter- 
mined by  the  intersection  of  any  three  surfaces  whatever, 
as  it  was  in  the  former  case  by  that  of  two  lines :  each 
of  these  three  surfaces  has,  in  like  manner,  all  its  condi- 
tions of  determination  constant,  excepting  one,  which 
gives  rise  to  the  corresponding  co-ordinates,  whose  pecu- 
liar geometrical  influence  is  thus  to  constrain  the  point 
to  be  situated  upon  that  surface. 

This  being  premised,  it  is  clear  that  if  the  three  co- 
ordinates of  a  point  are  entirely  independent  of  one  an- 
other, that  point  can  take  successively  all  possible  posi- 
tions in  space.  But  if  the  point  is  compelled  to  remain 
upon  a  certain  surface  defined  in  any  manner  whatever, 
then  two  co-ordinates  are  evidently  sufficient  for  deter- 
mining its  situation  at  each  instant,  since  the  proposed 
surface  will  take  the  place  of  the  condition  imposed  by 
the  third  co-ordinate.  We  must  then,  in  this  case,  un- 
der the  analytical  point  of  view,  necessarily  conceive  this 
last  co-ordinate  as  a  determinate  function  of  the  two 
others,  these  latter  remaining  perfectly  independent  of 
each  other.  Thus  there  will  be  a  certain  equation  be- 
tween the  three  variable  co-ordinates,  which  will  be  per- 


SURFACES.  253 

manent,  and  which  will  be  the  only  one,  in  order  to  cor- 
respond to  the  precise  degree  of  indetermination  in  the 
position  of  the  point. 

Expression  of  Surfaces  by  Equations.  This  equation, 
more  or  less  easy  to  be  discovered,  but  always  possible, 
will  be  the  analytical  definition  of  the  proposed  surface, 
since  it  must  be  verified  for  all  the  points  of  that  surface, 
and  for  them  alone.  If  the  surface  undergoes  any  change 
whatever,  even  a  simple  change  of  place,  the  equation 
must  undergo  a  more  or  less  serious  corresponding  mod- 
ification. In  a  word,  all  geometrical  phenomena  relating 
to  surfaces  will  admit  of  being  translated  by  certain  equiv- 
alent analytical  conditions  appropriate  to  equations  of 
three  variables ;  and  in  the  establishment  and  interpre- 
tation of  this  general  and  necessary  harmony  will  essen- 
tially consist  the  science  of  analytical  geometry  of  three 
dimensions. 

Expression  of  Equations  by  Surfaces.  Considering 
next  this  fundamental  conception  in  the  inverse  point  of 
view,  we  see  in  the  same  manner  that  every  equation  of 
three  variables  may,  in  general,  be  represented  geomet- 
rically by  a  determinate  surface,  primitively  defined  by 
the  very  characteristic  property,  that  the  co-ordinates  of 
all  its  points  always  retain  the  mutual  relation  enuncia- 
ted in  this  equation.  This  geometrical  locus  will  evi- 
dently change,  for  the  same  equation,  according  to  the 
system  of  co-ordinates  which  may  serve  for  the  construc- 
tion of  this  representation.  In  adopting,  for  example, 
the  rectilinear  system,  it  is  clear  that  in  the  equation  be- 
tween the  three  variables,  x,  y,  z,  every  particular  value 
attributed  to  z  will  give  an  equation  between  x  and  y,  the 
geometrical  locus  of  which  will  be  a  certain  line  situated 


254  MODERN  OR  ANALYTICAL  GEOMETRY. 

in  a  plane  parallel  to  the  plane  of  x  and  y,  and  at  a  dis- 
tance from  this  last  equal  to  the  value  of  z ;  so  that  the 
complete  geometrical  locus  will  present  itself  as  com- 
posed of  an  infinite  series  of  lines  superimposed  in  a  se- 
ries of  parallel  planes  (excepting  the  interruptions  which 
may  exist),  and  will  consequently  form  a  veritable  sur- 
face. It  would  be  the  same  in  considering  any  other  sys- 
tem of  co-ordinates,  although  the  geometrical  construction 
of  the  equation  becomes  more  difficult  to  follow. 

Such  is  the  elementary  conception,  the  complement  of 
the  original  idea  of  Descartes,  on  which  is  founded  gen- 
eral geometry  relative  to  surfaces.  It  would  be  useless 
to  take  up  here  directly  the  other  considerations  which 
have  been  above  indicated,  with  respect  to  lines,  and 
which  any  one  can  easily  extend  to  surfaces  ;  whether 
to  show  that  every  definition  of  a  surface  by  any  method 
of  generation  whatever  is  really  a  direct  equation  of  that 
surface  in  a  certain  system  of  co-ordinates,  or  to  deter- 
mine among  all  the  different  systems  of  possible  co-ordi- 
nates that  one  which  is  generally  the  most  convenient. 
I  will  only  add,  on  this  last  point,  that  the  necessary  supe- 
riority of  the  ordinary  rectilinear  system,  as  to  the  repre- 
sentation of  equations,  is  evidently  still  more  marked  in 
analytical  geometry  of  three  dimensions  than  fh  that  of 
two,  because  of  the  incomparably  greater  geometrical 
complication  which  would  result  from  the  choice  of  any 
.other  system.  This  can  be  verified  in  the  most  striking 
manner  by  considering  the  polar  system  in  particular, 
which  is  the  most  employed  after  the  ordinary  rectilinear 
system,  for  surfaces  as  well  as  for  plane  curves,  and  for 
the  same  reasons. 

In  order  to  complete  the  general  exposition  of  the  fun- 


CURVES    IN  SPACE.  255 

damental  conception  relative  to  the  analytical  study  of 
surfaces,  a  philosophical  examination  should  be  made  of 
a  final  improvement  of  the  highest  importance,  which 
Monge  has  introduced  into  the  very  elements  of  this  the- 
ory, for  the  classification  of  surfaces  in  natural  families, 
established  according  to  the  mode  of  generation,  and  ex- 
pressed algebraically  by  common  differential  equations,  or 
by  finite  equations  containing  arbitrary  functions. 

CURVES  OF  DOUBLE  CURVATURE. 

Let  us  now  consider  the  last  elementary  point  of  view 
of  analytical  geometry  of  three  dimensions  ;  that  relating 
to  the  algebraic  representation  of  curves  considered  in 
space,  in  the  most  general  manner.  In  continuing  to 
follow  the  principle  which  has  been  constantly  employed, 
that  of  the  degree  of  indetermination  of  the  geometrical 
locus,  corresponding  to  the  degree  of  independence  of  the 
variables,  it  is  evident,  as  a  general  principle,  that  when 
a  point  is  required  to  be  situated  upon  some  certain  curve, 
a  single  co-ordinate  is  enough  for  completely  determining 
its  position,  by  the  intersection  of  this  curve  with  the  sur- 
face which  results  from  this  co-ordinate.  Thus,  in  this 
case,  the  two  other  co-ordinates  of  the  point  must  be  con- 
ceived as  functions  necessarily  determinate  and  distinct 
from  the  first.  It  follows  that  every  line,  considered  in 
space,  is  then  represented  analytically,  no  longer  by  a 
single  equation,  but  by  the  system  of  two  equations  be- 
tween the  three  co-ordinates  of  any  one  of  its  points.  It 
is  clear,  indeed,  from  another  point  of  view,  that  since 
each  of  these  equations,  considered  separately,  expresses 
a  certain  surface,  their  combination  presents  the  proposed 
line  as  the  intersection  of  two  determinate  surfaces. 


256     MODERN   OR   ANALYTICAL   GEOMETRY. 

Such  is  the  most  general  manner  of  conceiving  the  alge- 
braic representation  of  a  line  in  analytical  geometry  of 
three  dimensions.  This  conception  is  commonly  consid- 
ered in  too  restricted  a  manner,  when  we  confine  our- 
selves to  considering  a  line  as  determined  by  the  system 
of  its  two  projections  upon  two  of  the  co-ordinate  planes  ; 
a  system  characterized,  analytically,  by  this  peculiarity, 
that  each  of  the  two  equations  of  the  line  then  contains 
only  two  of  the  three  co-ordinates,  instead  of  simulta- 
neously including  the  three  variables.  This  considera- 
tion, which  consists  in  regarding  the  line  as  the  intersec- 
tion of  two  cylindrical  surfaces  parallel  to  two  of  the 
three  axes  of  the  co-ordinates,  besides  the  inconvenience 
of  being  confined  to  the  ordinary  rectilinear  system,  has 
the  fault,  if  we  strictly  confine  ourselves  to  it,  of  intro- 
ducing useless  difficulties  into  the  analytical  representa- 
tion of  lines,  since  the  combination  of  these  two  cylin- 
ders would  evidently  not  be  always  the  most  suitable  for 
forming  the  equations  of  a  line.  Thus,  considering  this 
fundamental  notion  in  its  entire  generality,  it  will  be 
necessary  in  each  case  to  choose,  from  among  the  infinite 
number  of  couples  of  surfaces,  the  intersection  of  which 
might  produce  the  proposed  curve,  that  one  which  will 
lend  itself  the  best  to  the  establishment  of  equations,  as 
being  composed  of  the  best  known  surfaces.  Thus,  if 
the  problem  is  to  express  analytically  a  circle  in  space, 
it  will  evidently  be  preferable  to  consider  it  as  the  inter- 
section of  a  sphere  and  a  plane,  rather  than  as  proceed- 
ing from  any  other  combination  of  surfaces  which  could 
equally  produce  it. 

In  truth,  this  manner  of  conceiving  the  representation 
of  lines  by  equations,  in  analytical  geometry  of  three  di- 


CURVES   IN  SPACE.  257 

mensions,  produces,  by  its  nature,  a  necessary  inconve- 
nience, that  of  a  certain  analytical  confusion,  consisting 
in  this  :  that  the  same  line  may  thus  be  expressed,  with 
the  same  system  of  co-ordinates,  by  an  infinite  number 
of  different  couples  of  equations,  on  account  of  the  in- 
finite number  of  couples  of  surfaces  which  can  form  it ; 
a  circumstance  which  may  cause  some  difficulties  in  rec- 
ognizing this  line  under  all  the  algebraical  disguises  of 
which  it  admits.  But  there  exists  a  very  simple  method 
for  causing  this  inconvenience  to  disappear  ;  it  consists 
in  giving  up  the  facilities  which  result  from  this  variety 
of  geometrical  constructions.  It  suffices,  in  fact,  what- 
ever may  be  the  analytical  system  primitively  estab- 
lished for  a  certain  line,  to  be  able  to  deduce  from  it  the 
system  corresponding  to  a  single  couple  of  surfaces  uni- 
formly generated  ;  as,  for  example,  to  that  of  the  two 
cylindrical  surfaces  which  project  the  proposed  line  upon 
two  of  the  co-ordinate  planes ;  surfaces  which  will  evi- 
dently be  always  identical,  in  whatever  manner  the  line 
may  have  been  obtained,  and  which  will  not  vary  except 
when  that  line  itself  shall  change.  Now,  in  choosing 
this  fixed  system,  which  is  actually  the  most  simple,  we 
shall  generally  be  able  to  deduce  from  the  primitive  equa- 
tions those  which  correspond  to  them  in  this  special  con- 
struction, by  transforming  them,  by  two  successive  elim- 
inations, into  two  equations,  each  containing  only  two  of 
the  variable  co-ordinates,  and  thereby  corresponding  to 
the  two  surfaces  of  projection.  Such  is  really  the  prin- 
cipal destination  of  this  sort  of  geometrical  combination, 
which  thus  offers  to  us  an  invariable  and  certain  means 
of  recognizing  the  identity  of  lines  in  spite  of  the  diver- 
sity of  their  equations,  which  is  sometimes  very  greats 

R 


258     MODERN  OR  ANALYTICAL  GEOMETRY. 
IMPERFECTIONS   OF    ANALYTICAL    GEOMETRY. 

Having  now  considered  the  fundamental  conception  of 
analytical  geometry  under  its  principal  elementary  as- 
pects, it  is  proper,  in  order  to  make  the  sketch  complete, 
to  notice  here  the  general  imperfections  yet  presented  by 
this  conception  with  respect  to  both  geometry  and  to 
analysis. 

Relatively  to  geometry,  we  must  remark  that  the 
equations  are  as  yet  adapted  to  represent  only  entire 
geometrical  loci,  and  not  at  all  determinate  portions  of 
those  loci.  It  would,  however,  be  necessary,  in  some  cir- 
cumstances, to  be  able  to  express  analytically  a  part  of 
a  line  or  of  a  surface,  or  even  a  discontinuous  line  or 
surface,  composed  of  a  series  of  sections  belonging  to  dis- 
tinct geometrical  figures,  such  as  the  contour  of  a  poly- 
gon, or  the  surface  of  a  polyhedron.  Thermology,  es- 
pecially, often  gives  rise  to  such  considerations,  to  which 
our  present  analytical  geometry  is  necessarily  inapplica- 
ble. The  labours  of  M.  Fourier  on  discontinuous  func- 
tions have,  however,  begun  to  fill  up  this  great  gap,  and 
have  thereby  introduced  a  new  and  essential  improve- 
ment into  the  fundamental  conception  of  Descartes.  But 
this  manner  of  representing  heterogeneous  or  partial  fig- 
ures, being  founded  on  the  employment  of  trigonometri- 
cal series  proceeding  according  to  the  sines  of  an  infinite 
series  of  multiple  arcs,  or  on  the  use  of  certain  definite; 
integrals  equivalent  to  those  series,  and  the  general  in- 
tegral of  which  is  unknown,  presents  as  yet  too  much 
complication  to  admit  of  being  immediately  introduced 
into  the  system  of  analytical  geometry. 

Relatively  to  analysis,  we  must  begin  by  observing 


ITS   IMPERFECTIONS.  £59 

that  our  inability  to  conceive  a  geometrical  representation 
of  equations  containing  four,  five,  or  more  variables,  anal- 
ogous to  those  representations  which  all  equations  of  two 
or  of  three  variables  admit,  must  not  be  viewed  as  an  im- 
perfection of  our  system  of  analytical  geometry,  for  it 
evidently  belongs  to  the  very  nature  of  the  subject. 
Analysis  being  necessarily  more  general  than  geometry, 
since  it  relates  to  all  possible  phenomena,  it  would  be 
very  unphilosophical  to  desire  always  to  find  among  ge- 
ometrical phenomena  alone  a  concrete  representation  of 
all  the  laws  which  analysis  can  express. 

There  exists,  however,  another  imperfection  of  less 
importance,  which  must  really  be  viewed  as  proceeding 
from  the  manner  in  which  we  conceive  analytical  geom- 
etry. It  consists  in  the  evident  incompleteness  of  our 
present  representation  of  equations  of  two  or  of  three  va- 
riables by  lines  or  surfaces,  inasmuch  as  in  the  construc- 
tion of  the  geometric  locus  we  pay  regard  only  to  the 
real  solutions  of  equations,  without  at  all  noticing  any 
imaginary  solutions.  The  general  course  of  these  last 
should,  however,  by  its  nature,  be  quite  as  susceptible  as 
that  of  the  others  of  a  geometrical  representation.  It 
follows  from  this  omission  that  the  graphic  picture  of  the 
equation  is  constantly  imperfect,  and  sometimes  even  so 
much  so  that  there  is  no  geometric  representation  at  all 
when  the  equation  admits  of  only  imaginary  solutions. 
But,  even  in  this  last  case,  we  evidently  ought  to  be 
able  to  distinguish  between  equations  as  different  in 
themselves  as  these,  for  example, 

z!+3/*+l=0,  z6+y44-l=0,  ya+ex=0. 
We  know,  moreover,  that  this  principal  imperfection  of- 
ten brings  with  it,  in  analytical  geometry  of  two  or  of 


three  dimensions,  a  number  of  secondary  inconveniences, 
arising  from  several  analytical  modifications  not  corre- 
sponding to  any  geometrical  phenomena. 

Our  philosophical  exposition  of  the  fundamental  con- 
ception of  analytical  geometry  shows  us  clearly  that  this 
science  consists  essentially  in  determining  what  is  the 
general  analytical  expression  of  such  or  such  a  geomet- 
rical phenomenon  belonging  to  lines  or  to  surfaces  ;  and, 
reciprocally,  in  discovering  the  geometrical  interpretation 
of  such  or  such  an  analytical  consideration.  A  detailed 
examination  of  the  most  important  general  questions 
would  show  us  how  geometers  have  succeeded  in  actually 
establishing  this  beautiful  harmony,  and  in  thus  imprint- 
ing on  geometrical  science,  regarded  as  a  whole,  its  pres- 
ent eminently  perfect  character  of  rationality  and  of 
simplicity. 


Note. — The  author  devotes  the  two  following  chapters  of  his  course  to 
the  more  detailed  examination  of  Analytical  Geometry  of  two  and  of  three 
dimensions ;  but  his  subsequent  publication  of  a  separate  work  upon  this 
branch  of  mathematics  has  been  thought  to  render  unnecessary  the  repro- 
duction of  these  two  chapters  in  the  present  volume. 


THE   END. 


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